0 
0 

•;    0 

I  9 
3 
7 

1 
2 
3 


■      ARITHMETIC: 
1  HOW  TO  TEACH  IT 


A  MONOGRAPH 


BY 


FRANK  H.  HALL 


This  book  it  DUE  on  the  last  date  stamped  below 


S" 


This  book  is  DUE  on  the  last  date  stamped  below 

MAR  A     1827 

APR  ^1  1927 
viUN  4      1928 

FEB  3 -1958 
RECTQDMIisL 


MAR30J 


•n  L-9-15to-8,'26 


ARITHMETIC:  HOW  TO  TEACH  IT 


PROPOSED    CHANGES 


IN  THE 


METHODS  OF  TEACHING 
ARITHMETIC 

IN  THE  COMMON  SCHOOLS 


FRANK   H.   HALL 

Author  of  The  Werner  Arithmetics,  The  Hall  Arithmetics 
The  Arithmetic  Readers,  Etc. 


WERNER  SCHOOL  BOOK  COMPANY 

EDUCATIONAL  PUBLISHERS 
NEW  YORK  CHICAGO  BOSTON 


FROM   THE   REPORT 


OF  THE 


COMMITTEE  OF  TEN 


"The  Conference  [on  Mathematics] 
consisted  of  one  government  official  and 
university    professor,     five     professors    of 
mathematics    in    as    many   colleges,    one 
principal  of  a  high  school,  two   teachers 
of  mathematics  in  endowed  schools,  and 
one    proprietor    of   a   private    school  for 
boys.       The    professional    experience   of 
these  gentlemen  and  their  several  fields  of 
work  were  various,  and  they  came  from 
widely   separated    parts   of  the    country ; 
yet    they  were   unanimously   of   opinion 
that    a    radical   change  in  the  teaching    of 
arithmetic  was  necessary." 

Copyright,  1900 
Bv  WERNER  SCHOOL  BOOK  COMPANY 


Arithmetic  :    How  to  Teach  It 


>' 


QA 
\35 


ARITHMETIC:  HOW  TO  TEACH  IT. 


Whence  Comes  the  Demand  for  a  Change  in  the  Methods 
of  Instruction? 

The  "Committee  of  Ten"  was  appointed  at  the 
meeting  of  the  National  Educational  Association  in 
Saratoga  in  July,  1892.  Its  chairman  was  committee 
Charles  W.  Eliot,  President  of  Harvard  °*  ^en. 
University.  Dr.  William  T.  Harris,  Commissioner  of 
Education,  was  a  prominent  member  of  the  com- 
mittee. 

The  "Conference  on  Mathematics"   appointed  by 

\  this  committee  "consisted  of  one  government  official 

^  and  university  professor,  five  professors  of  mathematics 

,     in  as  many  colleges,  one  principal  of  a  high  school, 

*^    two  teachers  of  mathematics  in  endowed  schools,  and 

one  proprietor  of  a  private  school  for  boys,  conference  on 

The  professional  experiences  of  these  gen-  Mathematics. 

tlemen  and  their  several  fields  of  work  were  various, 

and  they  came  from  widely   separated  parts  of  the 

country;  yet  they  were  unanimously  of  opinion  that 

a  radical   change  in  the  teaching  of  arithmetic   was 

necessary."     The  members  of  this  conference  simply 

formulated  and  voiced  the  common  judgment  on  this 

subject — the  judgment  of  the  leaders  in  educational 

thought  and  of  intelligent  men  of  affairs  everywhere. 

5 


6  arithmetic:    how  to  teach  it 

McLellan  and  Dewey,  in  The  Psychology  of 
Number y  speak  of  "a  growing  impatience  with  the 
McLellan  meager  results  of  the  time  given  to  arith- 
«nd  Dewey,  nietic  in  the  traditional  course  of  the 
schools." 

Business  men  very  generally  deplore  the  lack  of 
ability  on  the  part  of  youthful  employees  fresh  from 
Business  ^^^  schools  to  figure  accurately.  Great 
Men.  emphasis  has  hitherto  been    given    to    the 

commercial  side  of  arithmetic,  and  yet  pupils  are 
inefficient,  a  majority  of  them  at  least,  to  a  remark- 
able degree,  in  the  figure  manipulation  required  in 
ordinary  business.  Said  a  prominent  Bostonian:  "All 
children  are  taught  to  cipher;  yet  in  my  selection  of 
office  boys  I  find  that  very  few  know  how  to  apply  the 
art  of  ciphering  to  the  work  of  life." 

High-school  teachers  are  wont  to  complain  of  the 
inefficiency  of  the  mathematical  training  received  by 
High-School  their  pupils  while  in  the  grades.  They 
Teachers.  ^^^  large  numbers  of  these  pupils,  possibly 
a  majority,  deficient  (i)  in  ability  to  discern  quantita- 
tive relation,  and  (2)  in  skill  in  accurate  ciphering. 
They  demand  improvement  in  these  two  respects. 

This  demand,  then,  for  a  radical  change  in  the 
method  and  curriculum,  so  far  as  arithmetic  is  con- 
cerned, is  of  no  narrow  origin.  It  is  as  extensive  as 
the  educational  world  itself,  and  as  intensive  as  the 
combined  expression  of  the  mathematician,  the  psy- 
chologist, the  representative  business  man,  and  the 
high-school  teacher  can  make  it. 


ARITHMETIC:      HOW   TO    TEACH    IT  7 

What  Changes  in  Matter  are  Desirable? 

The  first  recommendation  made  by  the  Conference 
on  Mathematics,  appointed  by  the  Committee  of 
Ten,  is,  that  "the  course  in  arithmetic  be  Abridgment 
at  the  same  time  abridged  and  enriched:  Necessary, 
abridged  by  omitting  entirely  those  subjects  which 
perplex  and  exhaust  the  pupil  without  affording  any 
really  valuable  mental  discipline;  and  enriched  by  a 
greater  number  of  exercises  in  simple  calculation  and 
in  the  solution  of  concrete  problems." 

This  Conference  suggested  the  curtailment  or  entire 
omission  of  compound  proportion,  cube  root,  obsolete 
denominate  quantities,  duodecimals,  and  the  -v^hat 
greater  part  of  commercial  arithmetic.  It  o™i"ed- 
further  suggested  that  in  such  subjects  as  profit  and 
loss,  bank  discount,  and  simple  and  compound  interest, 
examples  not  easily  made  intelligible  to  the  pupil 
should  be  omitted.  In  these  recommendations  the 
Conference  has  voiced  the  sentiment  of  thoughtful 
teachers  everywhere. 

Pupils  in  the  grammar  grades  have  been  required 
lo  memorize  definitions,  when,  because  of  the  imma- 
turity of  their  minds,  they  were  unable  to  see  through 
the  definitions  the  things  defined.  They  have  been 
required  to  solve  problems  of  whose  uses  and  applica- 
tions they  had  no  clear  conception.  Again  and  again 
they  have  lost  sight  of  what  they  were  trying  to  Jo  in 
trying  to  find  out  how  to  do  it.  The  mere  manipula- 
tion of  figures  has  been  allowed  to  absorb  their  atten- 
tion and  to  exhaust  their  energies, 

jfAT£NOi<l*l/iL  SCHOOL, 

liOS  HTlCEIiHS.  CPM. 


8  arithmetic:    how  to  teach  it 

In  many  instances  pupils  have  been  well  taught  in 
the  primary  grades  and  in  the  so-called  "mental  arith- 
metic" work.     They  have  learned  in  a  small 

Too  Much  .  ,      .  J 

Mere  Way    to  see    magnitude    relation    and    to 

Figure  Work,  gj^p^ggg  j^  j^  number.     But  no  sooner  are 

they  admitted  to  the  class  in  "written  arithmetic" 
than  they  are  confronted  with  long  rows  of  figures — 
to  them  mere  figures  —  and  with  these  they  are 
expected  to  juggle,  and  to  obtain  other  figures  called 
"the  answer." 

Says  General  Francis  A.  Walker:  "Who  of  us  has 
not  seen  in  the  hands  of  children  eleven,  twelve,  and 
Work  Too  thirteen  years  of  age  examples  in  com- 
Difficuit.  pound  and  complex  fractions  which  were 
more  difficult  than  any  operation  which  any  bank 
cashier  in  the  city  of  Boston  has  occasion  to  perform 
in  the  course  of  his  business  from  January  to  Decem- 
ber? The  most  jagged  fractions,  such  as  would  hardly 
ever  be  found  in  actual  business  operation — e.  g.,  ^^ 
or  ^ — are  piled  one  on  top  of  another  to  produce  an 
unreal  and  impossible  difficulty;  and  the  child,  having 
been  furnished  with  such  an  arithmetical  monstrosity, 
is  set  to  dividing  it  by  another  compound  and  complex 
fraction  as  unreal  and  ridiculous  as  itself.  All  this 
sort  of  thing  in  the  teaching  of  young  children  is  either 
useless  or  mischievous.  It  is  bad  psychology,  bad 
physiology,  and  bad  pedagogics." 

The  leading  educators  of  the  state  of  Wisconsin 
recently  recommended,  among  other  things, 
Wisconsin  the  following  in  relation  to  the  course  of 
¥duca  rs.      gtudy  in  arithmetic; 


ARITHMETIC:      HOW   TO   TEACH    IT  9 

Work  in  fractions  below  the  fifth  grade  mainly  oral. 

No  long  division  below  fifth  grade  with  divisors  of 
more  than  two  figures. 

Omit  greatest  common  divisor  entirely  as  separate 
topic. 

Omit  longitude  and  time.  Teach  the  principles  of 
this  in  connection  with  geography 

Omit  reduction,  addition,  subtraction,  multiplication, 
and  division  of  denominate  numbers  as  separate  topics. 

Limit  taxes,  insurance,  and  duties  to  simplest  cases 
and  explanation  of  terms. 

Give  very  little  attention  to  problems  in  interest. 

Omit  true  discount,  and  take  only  the  first  case  in 
bank  discount. 

Omit  cube  root  and  its  applications,  except  such  as 
can  be  done  by  inspection. 

From  the  foregoing  it  will  be  apparent  that  the  cur- 
rent of  thought  sets  strongly  in  favor  of  the  elimina- 
tion of  much  that  has  heretofore  been  regarded  as 
essential.  "For  ten  years,"  says  Superintendent 
J.  M.  Greenwood,  "the  process  of  elimination  has 
been  going  on,  and  we  have  not  seen  the  end  of  it  yet. ' ' 

Unmerited  Criticism  to  be  Expected  by  Those  Who 
Adopt  These  Recommendations. 

There  will  necessarily  be  some  embarrassment  for 
those  who  accept  and  adopt  such  abridgment  as  is 
herein  recommended.  There  are  examiners  and  ex- 
aminers. Many  of  these  learned  their  elementary 
mathematics  before  this  process  of  elimination  began, 
and  have  not  yet  come  into  line  with  those  educators 
who  favor  abridgment.  These  will  persist  in  con- 
fronting the  pupils  with  problems  that  belong  to  the 
parts  eliminated.     The  mechanically  taught  pupil  may 


lO  ARITHMETIC:      HOW    TO    TEACH    IT 

be  able  to  reduce  a  three-story  combination  of  com- 
pound and  complex  fractions  to  a  simple  fraction  in 
less  time  than  it  can  be  done  by  the  thoughtful  pupil. 
He  may  "get  answers"  to  problems  in  compound 
proportion  and  cube  root  and  its  applications  (if  the 
problems  are  in  every  way  regular  and  fall  under 
some  rule  that  has  been  memorized)  more  readily  than 
the  pupil  who  has  learned  to  see  magnitude  and  mag- 
nitude relation  in  his  figures — who  thinks.  Hence 
improved  methods  and  pupils  taught  in  accordance 
with  them  are  liable  to  unmerited  criticism  from  those 
whose  standard  is  radically  different  from  that  pre- 
sented by  the  Conference  on  Mathematics  appointed 
by  the  Committee  of  Ten. 

What  Enrichment  is  Desirable? 
The  work  must  be  enriched  as  well  as  abridged — 
"enriched  by  a  greater  number  of  exercises  in  simple 
„  calculation  and  in  the  solution  of  concrete 

More 

Concrete         problems."     The  pupil  must  be  exercised 

Work  •  • 

m  seemg  quantitative  relation.  The  em- 
phasis must  be  put  upon  this  phase  of  the  work  rather 
than  upon  mere  figuring.  The  figure  processes  must 
be  learned,  and  the  pupil  must  be  taught  to  compute 
with  absolute  accuracy.  But  from  the  first  he  must 
learn  to  regard  figure  manipulation  as  a  convenience 

„  .  .  .  in  discerning  exact  relation.  He  figures 
Cipnenng  to  °  ° 

be  Made  Less  not  that  he  may  learn  to  cipher,  but  that 

Prominent.        i  •   i  i  t  ,        , 

he  may  compare  quickly  and  accurately  the 
various  magnitudes  in  which  he  is  interested — mag- 
nitudes of  time,  of  space,  of  intensity,  of  weight,  of 


ARITHMETIC:     HOW   TO    TEACH   IT  II 

value.  Take  away  the  figures,  and  there  yet  remains 
mathematics  —  arithmetic  —  number.  Ciphering  will 
not  be  discarded  (though  it  would  be  possible  to  com- 
plete the  real  work  in  arithmetic  without  it),  but  it 
must  be  relegated  to  its  proper  place  in  mind  develop- 
ment. It  must  be  treated  as  a  convenience  in  obtain- 
ing mathematical  results. 

The  arithmetical  instruction  in  the  grades  must  be 
enriched,  just  as  the  teaching  of  reading  has  already 
been  enriched,  by  leading  the  pupils  to  see  through 
the  symbols  to  that  for  which  the  symbols  stand ;  by 
using  the  symbols  to  express  thought.  Thought  is 
the  main  thing  in  mathematics  as  well  as  in 

_  Magnitude 

reading.  In  either  of  these  branches  of  Relation  More 
study  one  may  play  with  symbols  or  work  ^°™'°®°*- 
with  symbols,  and  not  think.  "Thinking  is  discerning 
relation."  In  mathematics  the  things  related  are 
magnitudes.  Leave  these  out  —  juggle  with  mere 
figures — and  the  subject  is  impoverished;  put  them 
in,  and  the  subject  is  enriched. 

Number  Symbols  and  Their  Content. 

The  symbols  of  number  employed  in  arithmetic  are 
words,    as    one^    tzvo,    seven,    twenty-four,    etc. ;    and 
figures,  as  i,  2,  7,  24,  etc.      While  it  is  true  that  these 
symbols  stand  for  number  and  that  number  expresses 
ratio,  it  is  also  true  that  they  may  suggest  Double 
magnitude.     They  may   properly   stand  in  ^J^^i^er 
thought    for    measured    magnitudes.       For  Symbols, 
instance,   the   number  six  (or  the  figure  6)  may  call 
into    consciousness    a    rnagnitude    that    is    six    times 


12  arithmetic:    now  to  teach  it 

some  unit  of  measurement,  or  it  may  suggest  the  rela- 
tion (ratio)  of  that  magnitude  to  its  unit  of  measure- 
ment. The  following  from  The  Common  Sense  of  the 
Exact  Sciences*  will  help  to  make  this  plain : 

The  formulae  of  arithmetic  and  algebra  are  capable  of 
double  interpretation.  For  instance,  such  a  symbol  as  3 
meant,  in  the  first  place,  a  number  of  letters,  or  men,  or 
any  other  thing';  but  afterwards  was  regarded  as  mean- 
ing an  operation;  namely,  that  of  trebling  anything. 
And  so  the  symbol  \\  may  be  taken  either  as  meaning  so 
much  of  a  foot,  or  as  meaning  the  operation  by  which  a 
foot  is  changed  into  fifteen  inches. 

These  symbols,  then,  have  a  double  significance — 
a  possible  double  content.  Sometimes  they  should 
carry  with  them  the  thought  of  magnitude;  some- 
times the  thought  of  ratio.  Too  often  they  carry 
nothing.     They  are  empty. 

In  the  varying  content  of  these  number  symbols 
there  is  an  element  of  pedagogical  danger.  To  the 
Element  teacher  they  may  mean  the  one  thing;  to 
oi  Danger.  ^^vQ  pupil  the  Other,  or  possibly  nothing. 
He  figures,  but  he  does  not  see  relation.  He  ciphers, 
but  he  does  not  think.  How  can  he  think — discern 
relation — when  the  symbols  which  he  employs  are 
empty?  How  can  he  compare  when  the  terms  of  the 
comparison  are  not  in  consciousness — only  the  sym- 
bols of  the  terms? 

It  is  often  said  that  we  think  in  symbols.  So  we 
do;  but  they  are  symbols  of  something,  and  unless  we 
are  able  to  think  into  them  that  for  which  they  stand, 
our  thinking  is  not  of  the  highest  order.     To  make 

•W.  K.CIifford-D.  .^ppleton  &  Company. 


ARITHMETIC:      HOW   TO    TEACH    IT  1 3 

our  thinking  of  value,  we  must  be  able  to  interpret 
the  symbols  employed. 

When  Should  the  Number  Symbols  Suggest  Magnitude 
and  when  Suggest  Ratio? 

On   page   76   of    The  Psychology  of  Number,*   by 

McLellan  and  Dewey,  is  found  the  following: 

Psychologically  speaking,  can  the  multiplicand  ever 
be  a  pure  number?  If  the  foregoing  account  of  the  nature 
of  number  is  correct,  the  multiplicand,  however  written, 
must  always  be  understood  to  express  measured  quan- 
tity; it  is  always  concrete. 

But  if  multiplicands  are  always  concrete,  so  are 
products  and  addends  and  sums  and  minuends  and 
subtrahends  and  differences  and  remainders  pu^g  or 
and  dividends.  Multipliers  stand  for  rela-  concrete? 
tion,  or  as  Clifford  puts  it,  for  an  operation.  If 
a  particular  divisor  is  concrete,  the  corresponding 
quotient  shows  relation.  If  a  particular  divisor  shows 
relation,  the  corresponding  quotient  is  concrete. 

The  subject  of  arithmetic  will  be  marvelously 
enriched  when  the  number  symbols  bring  into  the  con- 
sciousness of  the  pupil  their  true  and  appropriate 
content. 

The  Imaging  of  Magnitude. 

The  magnitude  content  of  a  number  symbol  at  first 
will   be   a   memory    image  —  a   reproduction    of    that 
which  at  some  former  time  appeared  in  con-  Memory 
sciousness  through  the  action  of  the  senses.  i™a«es. 
It  will  be  an  image  of  some  cube  or  square  or  line  or 
circle,  that  was  presented  to  the  senses  at  some  par- 

♦D.  Appleton  &  Company. 


14  ARITHMETIC:     HOW    TO   TEACH    IT 

ticular  time  and  in  some  particular  place.  The  time 
and  the  place  will  at  first  be  recalled  as  well  as  the 
object.  .  - 

Reproducing  this  image  again  and  again,  using  it  in 
thinking — in  comparing — at  length  the  time  element 
and  the  place  element  fade  out,  and  it  can  be  made  to 
stand  forth  in  consciousness  as  a  cube,  a  square,  a  line, 
or  a  circle,  without  any  reference  to  any  particular 
Idealized  sense  magnitude.  Such  idealized  images 
Images.  make    up    a   stock    of    "mind    stuff"    with 

which  the  successful  pupil  builds  new  magnitudes  in 
endless  variety.  The  repeated  reproduction  of  mem- 
ory images  of  magnitude,  and  of  the  idealized  images 
to  which  these,  under  proper  instruction,  soon  give 
place,  is  the  sine  qua  non  to  a  proper  beginning  of 
number  teaching. 

But     again :     one     may     think    quantity    without 

thinking  any  particular /(3r;//  of  magnitude.     We  can 

think   4   as   a  magnitude  without  stopping 
Formless  2  b  rr     & 

Images  of  to  determine  whether  it  is  the  half  of  a  cube, 
Quan  ity.  ^  square,  a  line,  or  a  circle.  We  may  thus, 
without  being  tied  down  to  the  forms  of  sense,  dis- 
criminate sharply  between  the  one-half  whose  prac- 
tical content  is  magnitude  and  the  one-half  whose 
content  is  relation.  We  can  thus  put  content  into 
number  symbols,  even  though  our  images  of  magni- 
tude are  without  definite  shape.  We  can  even  think — 
see  magnitude  relation  clearly  —  when  the  things 
related  are  to  us  literally  "without  form  and  void." 
So  we  learn  to  deal  with  abstract  magnitude — abstract 
quantity. 


ARITHMETIC:     HOW   TO   TEACH    IT  1$ 

Yet  again :  one  can  think  two  number  symbols 
of  dissimilar  content,  and  a  suggested  operation, 
without  stopping  to  think  which  of  the 
number  symbols  has  a  magnitude  content  with 
and  which  a  ratio  content.  The  equation,  y™°^- 
|.X^=5,  is  true,  whether  the  ^  or  the  -^  is  regarded 
as  a  symbol  of  magnitude.  It  is  not  necessary  in  prac- 
tice that  one  should  think  which  of  these  stands  for 
magnitude  and  which  stands  for  ratio;  but  it  is  neces- 
sary that  in  mind  training  the  teacher  should  provide 
that  the  pupil  shall  not  as  a  rule  use  symbols  and  pro- 
cesses which  he,  the  learner,  the  one  who  is  being 
trained,  is  unable  to  interpret.  "We  may  always 
depend  upon  it,"  says  W.  K.  Clifford,  "that  algebra 
which  cannot  be  translated  into  good  English  and 
sound  common  sense  is  bad  algebra."     So 

^  Ability  to 

arithmetic  whose  expressions  and  processes  Translate 
cannot  be  translated  by  the  learner  himself     ^  «*^an^- 
into  magnitudes  and  magnitude  relation  is  bad  arith- 
metic— bad  pedagogy. 

Number  and  Measurement. 

The  number  idea  has  its  origin  in  measurement .  In 
The  Psychology  of  Number*  (McLellan  and  Dewey), 
page  44,  the  authors  speak  of  "the  process  of  measur- 
ing from  which  number  has  its  genesis." 

The  teaching  of  number  begins  in  measurement . 
The  very  best  teachers  in  our  primary  grades  now 
accept  this  as  a  fundamental  truth,  and  base  their 
practice  upon  it. 

*D.  Appleton  &  Company.  -'Jlftjili 


l6  ARITHMETIC:     HOW  TO   TEACH   IT 

The  uses  of  number  end  in  measurement.  This 
is  in  accord  with  our  daily  experience  and  observa- 
tion. We  learn  arithmetic  that  we  may  measure — 
measure  our  wealth;  measure  the  land  and  its  prod- 
ucts; measure  the  heights  of  mountains  and  the 
depths  of  the  sea;  measure  heat,  light,  and  the  elec- 
tric current. 

Hence,  in  the  beginning,  arithmetic  deals  with 
magnitudes — something  to  measure.  In  the  end  it 
deals  with  magnitudes — something  to  be  measured. 
But  in  the  middle  there  has  been  in  the  past  a  great 
gulf  of  figures  and  figure  processes.  Many 
Figure  a  pupil  has  been  figuratively  shipwrecked  in 

Processes.  crossing  this  gulf.  The  small  minority  of 
mentally  strong  pupils  make  the  passage  without  seri- 
ous disaster.  It  is  just  possible  that  the  very  few,  the 
exceptionally  strong  ones,  are  even  made  stronger  by 
the  difficulties  which  they  encounter.  These  are  the 
mathematicians.  They  will  learn  mathematics,  what- 
ever may  be  the  method  of  presentation.  But  to  the 
great  majority  it  is  an  unfortunate  experience  from 
which  they  never  fully  recover.  The  new  methods 
demand  the  bridging  of  this  gulf  between  the  magni- 
tude measurements  in  which  the  number  idea  origi- 
nates and  the  magnitude  measurements  to  which  the 
arithmetical  processes  are  to  be  applied;  or  better, 
perhaps,  the  discovery  of  the  magnitude  islands  that 
mark  a  passage  (possibly  a  little  circuitous)  across  the 
gulf.  We  have  no  right  to  allow  the  youthful  mariner 
to  lose  sight  of  land  for  any  considerable  time  while 
making  this  trip. 


ARITHMETIC:      HOW    TO    TEACH    IT  1/ 

The  figure  processes  he  must  learn ;  but  to  require 
him,  to  allow  him,  to  abandon  thought  of  magnitude 
while  he  learns  to  figure  has  its  parallel  in  Must  not 
the  study  of  the  words  of  a  sentence  with-  i^^^^^^  ^^ 
out  any  attention  to  the  thought  expressed.  Magnitude. 
Thought  is  the  main  thing  in  arithmetic  as  well  as  in 
language.  Figures  are  at  once  the  symbols  of  the  real 
subjects  of  thought  and  of  the  relations  of  these  sub- 
jects. To  allow  the  pupil  to  juggle  with  these  sym- 
bols for  weeks  and  months  while  magnitude  is  either 
altogether  omitted  or  thrust  far  into  the  background 
is  fatal  to  good  training.  To  teach  a  child  long  divi- 
sion and  long  multiplication  and  long  fraction  manipu- 
lation before  he  is  made  to  feel  the  need  of  these 
processes  in  making  measurements  and  in  seeing  mag- 
nitude relations,  is  not  simply  a  waste  of  time:  it  is 
unpedagogical,  and  in  many  instances  at  least, 
seriously  disastrous  as  an  attempted  step  in  mental 
development. 

A  Greater  Degree  of  Accuracy  in  the  Figure  Processes 
Demanded. 

The  Conference  on  Mathematics  appointed  by  the 
Committee  of  Ten  summed  up  the  suggestions  made 
in  their  special  report  on  arithmetic  under  two  heads — 
namely,  (i)  The  giving  of  the  teaching  a  more  concrete 
form,  and  (2)  The  paying  of  more  attention  to  facility 
and  correctness  in  work. 

The  first  of  these  suggestions  has  already  been 
considered.  Taking  care  that  the  pupil  associates 
thoughts  of  magnitude  with  figures  and   figure  pro- 


1 8  ARITHMETIC:     HOW   TO   TEACH    IT 

cesses    is    giving    the    teaching    "a    more    concrete 
form." 

It  now  remains  for  us  to  consider  the  second  sug- 
gestion— the  securing  of  facility  and  correctness  in  the 
number  processes;  Heretofore  the  principal  part  of 
the  work  in  "written  arithmetic"  has  been  ciphering, 
and  yet  a  high  degree  of  accuracy  has  not  been  secured. 
(Facility  must  not  for  a  moment  be  considered  apart 
from  accuracy.)  If  we  now  devote  less  time  to  the 
mere  figure  processes  and  more  time  to  the  discerning 
of  magnitude  relation,  may  not  the  so-called  improved 
methods  result  in  even  greater  inaccuracy,  and  conse- 
quently in  diminished  practical  efficiency? 

How  Secure  Greater  Accuracy? 

The  degree  of  approach  to  accuracy  by  a  pupil 
does  not  depend  so  much  upon  the  amount  as  upon 
the  character  of  the  work  done.  Careless  facility  is 
not  merely  useless:  it  is  positively  harmful.  Hence, 
while  the  problems  provided  for  the  pupil  may  well 
be  much  more  simple  in  respect  to  the  amount  of 
figuring  required,  the  importance  of  accuracy  must  be 
emphasized  to  a  very  much  greater  degree  than  has 
Inaccurate  usually  been  the  custom  of  teachers  in  the 
^t'be""**  grades.  Indeed,  the  pupil  must  not  be  com- 
commended.  mended  at  all  for  inaccurate  work — for  work 
in  which  there  is  one  wrong  figure!  It  must  be 
impressed  upon  him  in  the  very  beginning  that  cipher- 
ing in  which  there  are  errors  has  no  value  whatever. 
His  task  must  be,  not  the  solution  of  ten  problems 
with  but  few  errors,  but  rather  as  many  problems  as 


ARITHMETIC:      HOW    TO   TEACH    IT  I9 

he  can  solve  without  making  any  mistakes.      His  seat 
work  in  arithmetic  (and  his  home  work,  too,  if  any 
be  assigned)  should  be,   for  the  most  part,   „. 
mechanical,  and  so  simple  that  he  can  con-  seat  work 
centrate  his  whole  energy  upon  the  matter 
of  accuracy.      It  should  be   something  that  he    well 
knows  how  to  do,  the  only  question  being,  Can  he  do 
it  accurately?     In  this  way,  and  in  this  way  only,  can 
proper  emphasis  be  put  upon  the  importance  of  abso- 
lute correctness. 

Marking  Arithmetic  Papers. 

When  papers  (or  slates)  upon  which  is  the  work  of 
many  pupils  to  whom  the  task  of  copying  and  figuring 
had  been  assigned,  are  presented  to  the  teacher  for 
examination,  it  is  not  well  for  her  to  consider  too 
much  the  number  of  errors  made  by  each  pupil.  Each 
paper  is  right  or  wrong;  perfect  or  imper-  « 'Perfect"  01 
feet;  good  or  worthless.  Whether  it  con-  "imperfect.' 
tains  one  error  or  ten,  it  must  be  put  into  the  im- 
perfect class.  All  the  pupils  who  make  mistakes  in 
figuring  must,  for  the  moment  at  least,  be  classed 
together,  whether  the  number  of  errors  is  two  or  ten. 
In  either  case  the  work  is  unsatisfactory,  unreliable, 
worthless. 

In  the  opinion  of  the  writer,  if,  in  the  daily  tests 
of  the  ability  of  pupils  in  figuring,  more  than  twenty- 
five  per  cent  of  the  papers  are  imperfect,  the  Teacher  may 
teacher  is  at  fault.  Either  the  lesson  is  too  ^  **  ^*'^^- 
heavy,  or  the  teacher  does  not  sufficiently  impress  upon 
the  pupils  the  importance  of  accuracy  in  ciphering. 


20  ARITHMETIC:     HOW   TO   TEACH    IT 

The  seat  work,  the  mere  practice  in  figuring,  should 
be  made  so  light,  and  the  pupils  encouraged  to  exer- 
cise so  much  care  in  the  doing  of  it,  that 

The  Degree  ° 

of  Accuracy  seventy-five  to  ninety  per  cent  of  the  papers 
Expected.       ^jjj    ^^   perfect.       When    this    degree    of 

accuracy  has  been  attained,  the  amount  of  daily  work 
/or  those  pupils  who  usually  present  perfect  papers, 
may  be  somewhat  increased ;  but  in  all  cases  and  in  all 
the  grades,  infallible  accuracy  must  be  the  aim.  At 
most,  nothing  beyond  the  first  error  should  be  counted 
to  the  credit  of  the  pupil.  To  what  length  can  the 
pupil  continue  to  manipulate  figures  without  one  error? 
is  the  question  for  the  examiner  and  for  the  pupil. 

Old  Method  of  Marking  Papers. 

Too  often  it  has  been  the  custom  to  mark  a  paper 
90  if  only  one  problem  in  ten  contains  an  error. 
Often — shall  I  say  usually? — the  pupil  has  been  taught 
to  believe  that  90  per  cent  of  accuracy  in  the  third 
grade  is  good.  If  only  one  figure  was  wrong,  the 
paper  was  marked  95,  and  95  is  excellent.  This  has 
been  the  method  of  marking,  too,  in  the  fourth 
grade  and  in  the  fifth  grade,  and  in  all  the  grades  up 
to  and  including  the  eighth.  Then  perhaps  the  pupil 
leaves  school.  For  six  years  he  has  been  taught 
that  95  in  figure  processes  is  excellent;  90,  good;  80, 
Ninety  fair,  and  even  70  good  enough  to  "pass." 

in  Accuracy!  ^^  goQ.s>  out  into  the  business  world,  to 
is  Failure.  *  learn  that  90  per  cent  of  accuracy  in  figur- 
ing, instead  of  being  good,  is  absolute  failure;^)  that 
there  is  no  place  in  the  world  for  a  ninety-per-cent 


ARITHMETIC:     HOW  TO   TEACH    IT  21 

accountant.  His  inaccurate  facility  in  the  use  of  figures 
gained  for  him  much  credit  in  the  schoolroom,  but 
in  the  store  it  is  worthless.  The  fact  that  he  knows 
how  to  solve  the  problems,  and  can  explain  them  with 
the  "hences"  and  "sinces"  in  their  proper  places,  is 
of  no  avail  in  his  effort  to  retain  his  place  as  an 
accountant.  He  is  inaccurate;  hence  his  work  is  of 
no  value  whatever. 

A  nearer  approach  to  accuracy  may  be  made,  not  by 
a  greater  amount  of  careless  manipulation  of  figures  in 
difficult  problems,  but  by  the  careful  solution  of  many 
simple  problems  in  which  the  principal  effort  on  the 
part  of  both  teacher  and  pupil  is  to  secure  results  that 
are  correct  in  every    respect.      The   most    important 
part  of  the  work  of  the  teacher  in  this  effort  is  not  the 
correcting   of   the  pupil's  mistakes:    it   is   rather  the 
training  of  the  pupil  into  such  careful  habits  Prevention 
that   mistakes    will  not    be    made.  (Many  BeSrThan 
a  teacher  sits  up  at  night  to  correct  errors  correction, 
that    she    might    better   sit    up    in    the    daytime   to 
prevent.  J 

Number  Facts  to  be  Memorized. 

The  primary  number  facts  must  be  perfectly  mem- 
orzied.  ^Perceive,  express,  MEMORIZED  is  the  order  in 
which  the  work  must  be  done.  In  days  gone  by 
pupils  were  sometimes  required  to  memorize  and 
express  that  which  they  did  not  perceive.  We  must 
not  now  allow  them  simply  to  perceive  and  express 
that  which  they  ought  to  memorize. 


22  ARtTHMETiC:     HOW  TO   TEACH   IT 

Amount  to  be  Memorized. 

The  amount  to  be  memorzied  is  not  appalling. 
One  new  important  primary  number  fact  learned  each 
day  from  the  time  the  child  enters  the  third  grade 
One  Fact  until  he  enters  the  sixth  will  put  into  his 
Bach  Day.  possession  a  stock  of  mathematical  "mem- 
ory stuff"  that  will  compare  favorably  with  that  pos- 
sessed by  the  average  eighth-grade  pupil  of  to-day. 
The  imperfect  memory  work  in  many  of  our  schools 
may  be  attributed  to  the  facts,  (i)  that  the  advance- 
ment is  not  along  definite  lines — it  is  haphazard ;  and 
(2)  that  the  work  of  the  individual  pupil  is  not  closely 
enough  observed  and  directed. 

Let   us   make  a  brief  survey  of  the   field  for  these 

memory    operations.       There  are    forty-five    primary 

facts  of  addition.  (See  Werner  Arithmetic, 
Forty-flve  ^         A 

Facts  of         Book  II.,  page  273).*^,  These  facts  properly 

memorized  will  carry  with  them  the  primary 
facts  of  subtraction.  The  boy  who  has  perceived, 
expressed,  and  memorized  the  fact  that  4  and  3  are 
7  (' ' ' '  ' ' ' )  cannot  fail  to  know  that  7  less  3  are  4, 
and  that  7  less  4  are  3. 

There  are  sixty-four  primary  facts  of  multiplication. 
(See  Werner  Arithmetic,  Book  II.,  note  at  bottom  of 
Sixty-four  P^^^  ^^4.  and  tables  on  page  275.)!  These 
Facts  of  Mui-  facts  properly  memorized  will  carry  with 
^  °°'  them  twice  sixty-four  primary  facts  of  divi- 
sion. The  boy  who  has  perceived,  expressed,  and  mem- 
orized the  fact  that  four  fives  equal  20  (" " '  "'"  "  '" 

•See  Complete  Arithmetic,  page  443. 
fSee  Complete  Arithmetic,  page  444,  note  6. 


ARITHMETIC:      HOW    TO    TEACH    IT  23 

""')  will  also  know  that  5  is  contained  in  20  four 
times,  and  that  ^  of  20  is  5.  There  are,  then,  but  one 
hundred  nine  (45+64)  primary  facts  of  number  to  be 
memorized  in  order  that  the  pupil  may  be  prepared 
to  cipher  in  the  fundamental  processes.  At  least 
forty-five  of  these  (33  facts  of  addition  and  12  of  mul- 
tiplication) will  usually  be  learned  by  the  pupil  before  he 
enters  the  third  grade.  This  leaves  but  sixty-four  of  the 
above  to  be  memorized  after  entering  the  third  grade — 
and  there  are  nearly  two  hundred  days  in  a  school  year! 

Individual  Work  to  be  Done. 

If,  at  the  beginning  of  the  third-grade  work,  the 
teacher  would  assist  each  pupil  in  taking  an  accurate 
inventory  of  his  memory  stock,  and  would  inventory 
then  take  care  that  a  little  is  added  to  it  *<»  ^«  t^"^**"- 
each  day,  the  memory  side  of  the  task  mathematical 
would  not  be  a  formidable  one.  In  this  work,  as  well 
as  in  imaging  magnitude,  the  teacher  must  know  as 
nearly  as  possible  the  exact  mental  status  of  each 
pupil.  She  must  not  lose  sight  of  the  individual  in 
the  class.  She  must  not  attempt  to  have  her  pupils 
memorize  by  platoons. 

Surely,  at  the  end  of  the  third  school  year  the  pupil 
should  have  complete  memory  possession  of  the  one 
hundred  nine  facts  mentioned  above.  Moreover,  he 
should  be  able  to  use  them  accurately  in  easy  exam- 
ples. Make  the  ciphering  work  very  light,  and  insist 
upon  perfect  accuracy,  should  be  the  teach-  The  Rule, 
er's  inflexible  rule  in  this  work  and  in  the  grade  above. 
There  should  be  no  occasion  for  the  criticism  that 


24  ARITHMETIC:      HOW    TO    TEACH    IT 

"the  multiplication  table  is  neglected,"  and  that  "the 
pupils  are  interested  in  their  work  and  seem  very  bright, 
but  they  are  not  accurate  in  the  figure  processes," 

The  Denominate  Number  Tables. 

The  pupil  should  be  made  familiar  with  the  com- 
mon units  of  measurement.  But  this  cannot  be 
accomplished  by  merely  memorizing  the  words  and 
figures  of  the  tables.  So  far  as  possible  he  should  use 
Weights  and  these  units  in  actual  measuring.  Sets  of 
Measures.  weights  and  measures  are  even  more  neces- 
sary in  the  equipment  of  the  schoolroom  than  black- 
board and  crayon. 

Some  of  these  units  of  measurement  should  be  pre- 
sented to  the  pupil  at  the  very  beginning  of  the  work 
in  arithmetic — even  before  its  formal  introduction  as 
one  of  the  daily  subjects  in  the  school  curriculum. 
Many  a  boy  excels  in  arithmetic  because  of  a  home 

environment  by  which  he  was  early  led   to 
Measuring  •'  ■' 

lor  a  measure  for  a  purpose.      Perhaps  he  meas- 

^^^******  ured  to  make,  or  measured  to  sell,  or  meas- 
ured in  play.  If  he  measured,  he  dealt  with  magni- 
tudes, and  was  thus  learning  to  consider  simple,  exact 
magnitude  relations,  and  to  express  them  numeri- 
cally. He  made  the  best  possible  beginning  in  arith- 
metic at  home — in  measurement. 

The  home  surroundings  of  many  pupils  are  not 
favorable  to  such  foundation  laying;  hence  the  means 
for  this  must  be  provided  at  school.  It  is  useless,  it 
is  positively  harmful,  to  attempt  the  work  in  arithme- 
tic without  it. 


ARITHMETIC:      HOW   TO   TEACH    IT  2$ 

The  facts  usually  presented  in  the  denominate 
number  tables  must  not  be  kept  from  the  pupil  until 
he  has  learned  how  to  "cipher  in  simple  numbers  and 
in  fractions,"  and  then  given  in  doses  of  one  or  two 
tables  a  day!  Such  facts  are  an  essential  part  of  the 
mathematical  foundation,  and  as  such  must  „ 

Measure 
be  introduced  in  the  beginning  of  the  work,   in  the 

The  pupil  may  be  led  to  "measure  for  con-     ^^^^^^s- 
structive  purposes"  or  for  destructive  purposes.     He 
may  measure  to  buy,  or  measure  to  sell,  or  measure 
in  games;  but  measure  he  must,  if  he  is  expected  to  dis- 
cern magnitude  relation  and  to  express  it  numerically. 
The  pupil  should  early  become  familiar  with  the 
terms  tnc/i,  foot,  square  inch,  square  foot,  cubic  inch, 
cubic  foot,  pound,  ounce,  pint,  quart,  gallon, 
etc.    These  terms  should  bring  into  his  con-  suggest 
sciousness  the  magnitudes  for  which  they       ^^  "  ^^* 
stand,  not  necessarily  the  tables  in  which  the  words 
are  found. 

Reviews  Necessary. 

So  frequently,  moreover,  must  the  pupil  make  use  of 
the  number  facts  that  are  usually  given  in  the  denom- 
inate number  tables,  and  so  systematically  must  he  be 

led  to  review  (see  again)  these  facts,  that  „   ^  „ 

\  o       J  jjlust  Become 

they  will  at  length  become  to  him  a  perma-  a  Permanent 

,  •  tr      Possession. 

nent   memory   possession.      He  may  never 

be  asked,  perhaps,  to  commit  to  memory  the  "tables" 
as  of  yore,  but  the  facts  therein  contained  will  be 
memorized  before  he  reaches  that  part  of  the  text- 
book in  which  the  "tables"  are  usually  given. 


96  ARITHMETIC:     HOW    TO   TEACH    IT 

The  Time  to  Begin  the  Formal  Work  in  Arithmetic. 

Since  the  number  idea  originates  in  measurement, 

and  since  knowledge  and  skill  in  arithmetic  are  acquired 

for  purposes  of  measurement,  it  is  not  advisable  to 

push  to  the  front  the  mere  figure  processes  in  the  early 

stagfes  of  the  school  work.      Only  as  the 

Figures  to  be  °  •' 

Kept  in  the  pupil  IS  made  to  feel  the  need  of  number 
Background.  ^^^^  figures)  in  discerning  and  expressing 
quantity  relation  should  the  attempt  be  made  to  enlist 
his  interest  in  such  exercise.  True,  it  is  possible  to 
force  the  figure  processes  upon  the  attention  of  the 
pupil  at  a  very  early  age,  and  to  secure  seemingly 
excellent  results.  But  it  is  not  desirable  to  do  this. 
Other  branches  of  study,  if  not  more  important,  much 
better  adapted  to  the  needs  of  the  young  learner, 
should  absorb  the  principal  part  of  his  attention  in 
„^  „,  ^       the  first  two  years  of  school  life.     While 

The  First  ,  ^ 

Two  Years  of  engaged   in   the  study  of  his  own  environ- 

School  Life.  i.       •       i  •         ^  .        j  • 

ment;    m   learnmg  to  compare,   to  discern 

relation,  to  think;  and  to  express  his  thought  in  lan- 
guage, in  drawing,  in  making;  in  learning  to  read  and 
to  write — he  incidentally  becomes  familiar  with  such 
magnitudes  and  measurements  as  will  make  the  only 
possible  foundation  for  sound  mathematical  reasoning. 
Figures  and  figure  processes  should  be  kept  in  the 
background,  and  called  into  prominence  only  as  their 
necessity  is  felt  by  the  pupil  in  his  efforts  to  under- 
stand his  environment  and  to  solve  the  child  problems 
that  naturally  confront  him. 

When  this  course  is  pursued,  and  formal  arithnietic 


J 


ARITHMETIC:     HOW   TO   TEACH    IT  2^ 

work  put  off  until  the  last  part  of  the  second  school 
year,  or  even  to  the  beginning  of  the  third,   Formal 
more  will  be  accomplished  in  a  single  term  ^"rk^rbe 
of  ten  or  twelve  weeks  than  would  other-  Deferred, 
wise  be  accomplished  in  twice  as  many  months. 

How  to  Lay  the  Foundation. 

In  the  early  years  the  arithmetical  foundation  may 
be — ought  to  be — laid  in  connection  with  the  work 
in  drawing,  in  nature  study,  in  games,  and  in 
construction  work  of  all  kinds.      Even  if  the  Nature 
teacher  is  not  thoughtful  enough  to  see  that      "  y»  ^  <'• 
this  is  done,  the  deferring  of  the   .ormal  arithmetic 
work  until  the  beginning  of  the  third  school  year  will 
give  opportunity  for  the  child   to  secure  much  of  this 
foundation  materialin  the  home,  in  the  store,  on  the 
farm,  in  the  workshop,  on  the  playground — anywhere 
and  everywhere  that  he  finds  something  to  be  measured. 

Hence  it  is  to  the  advantage  not  only  of  the  child 
who  is  being  well  trained  in  school,  but  of  the  child 
who   is    taught    mechanically   and    unpeda- 

°  J  ^  Bad  Teaching 

gogically,  that  the  formal  work  in  number  worse  than 
should  not  be  commenced  in  the  first  years  °  ^^'^  '°^' 
of  school.  A  little  good  instruction  will  do  no  harm ; 
but  bad  number  teaching  in  these  years  is  worse  by 
far  than  no  teaching  at  all.  Too  much  good  teaching 
in  the  discerning  of  quantitative  relation  will  too  Much 
result  inevitably  in  the  neglect  of  something  ?°°^  Teach- 

J  o  t>    ing  may  be 

more  important,    and   may   lead   the    child  Harmful, 
forever  after  to  put  too  much  emphasis  upon  the  one 
subject  of  magnitude  and  magnitude  relation. 


28  ARITHMETIC:     HOW  TO  TEACH   IT 

How  to  Begin  the  Formal  Number  Work. 

If  the  beginning  of  the  regular  daily  work  in  arith- 
metic is  deferred  until  the  pupils  are  seven  or  eight 
years  of  age,  they  will  have  acquired  incidentally  many 
of  the  number  facts  given  on  pages  6  and  7  of  Book 
I.  of  the  Werner  Arithmetics.*  The  teacher  should 
study  now    "take   an    account    of   stock."       She 

Each  Pupil,  should  study  each  pupil  in  respect  to  his 
imaging  power  and  his  memory  acquirements. 

If  the  pupil  has  been  trained  properly,  he  will  be 
able  easily  to  call  into  consciousness  such  magnitudes 
as  are  suggested  by  the  following  words:  inch,  six 
inches,  foot,  yard,  pint,  quart,  square,  i-inch  square, 
2-inch  square,  cube,  i-inch  cube,  2 -inch  cube,  a  half -inch. 
Imaging  ^  half-foot,  one  third  of  a  foot,  two  thirds  of 
Po'^er*  a  foot,  a  half -hour,  a  quarter  of  an  hour,  a 

half-dollar^  "«  quarter,''  a  dime,  five  cents,  ten  cents, 
a  dozen,  half  a  dozen,  etc.  Whatever  his  training  or 
his  lack  of  training  may  have  been,  he  will  be  familiar 
with  some  of  these  terms,  and  with  many  expressions 
of  quantity  and  quantity  relation  not  here  given. 

He  will  have  memorized  some  number  facts,  such 

as  2   fives  are   10,  2   twos  are  4,  2  "quarters"  equal 

a  half-dollar,    15   minutes  and    15    minutes 

Memory  ■'  •' 

of  Number  are  a  half-hour.  Perhaps  he  knows  that 
Facts,  ^   ^ggg  ^^^   ^   ^ggg   ^^^   ^   eggs,   and   that 

6  eggs  and  6  eggs  are  12  eggs;  that  i  half  of  a  pie  and 
I  fourth  of  a  pie  are  3  fourths  of  a  pie;  that  i  half  of 
a  pie  and  i  sixth  of  a  pie  are  4  sixths  of  a  pie ;  that 

*  Substantially  the  same  tacts  are  given  on  patres  S  and  6  of  Hall's  Elementary 
Arithmetic. 


ARITHMETIC:      HOW   TO   TEACH    IT  29 

a  half-dollar  and  a  "quarter"  are  together  equal  in 
value  to  75  cents;  that  a  lO-spot  is  made  up  of 
2  fours  and  2 ;  that  a  7-spot  is  made  up  of  2  threes 
and  I  ;  that  a  half  of  3  feet  is  i  and  i  half  feet ;  that 
2  yards  equal  6  feet;  that  10  tens  equal  100;  that  half 
of    100   is    50;  that   half   of   500  is  250;  that  half  of 

5  dollars  is  2  and  i  half  dollars;  that  it  takes  4  i-inch 
squares  to  make  a  2-inch  square,  and  8  i-inch  cubes 
to  make  a  2-inch  cube,  etc. 

Werner  Arithmetic,  Book  I.,  Pages  6  and  7.* 

The  teacher,  having  gained  definite  knowledge  of 
the  mental  attainments  of  each  pupil  so  far  as  number 
is  concerned,  should   now  proceed  to  teach 

^    ^  Teach  New 

such  of  the  number  facts  given  on   pages  Number 

6  and  7  of  Book  I.  as  have  not  already  been 
learned.     With  some  pupils  this  will  be  a  slight  task. 
With   others    much    patient  effort  will  be    required. 
With  pupils  who  easily  image  the  magnitudes  consid- 
ered, almost  the  entire  strength  may  be  concentrated 
upon  the  memory  phase  of  the  work,  and  the  task 
will  be  speedily  and  well  done.     To  pursue  a  seem- 
ingly similar  course  with  pupils  who  do   not   image 
easily  will  give  mere  word   memory  results.   Mere  word 
The  number  of  such  pupils  will  be  relatively  R^™°^un- 
fewer  than    it  would    have  been   had    the  satisfactory, 
formal  number  work  been  begun  in  the  first  grade. 
But  the  teacher  must  at  all  times  be  on  her  guard  lest 
she   be   deceived   by   the   false  testimony   of  a  good 
memory  of  words  and  sentences. 

♦Hall's  Elementary  Arithmetic,  pages  5  and  6. 


30  ARITHMETIC:      HOW    TO   TEACH    IT 

Here,  then,  is  opportunity  for  most  important  and 

practically  valuable    child-study.     The    teacher    will 

need  all  the  skill  she  can  command  to  enable 

The  Real 

Content  of  the  her  to  determine  the    real  content  of    the 

Child  Mind,  child's  mind  when  he  says  "2  thirds  of  6  are 
4";  "i  half  of  6  is  3";  "2  quarts  are  4  pints"; 
"3  quarts  are  6  pints,"  etc.  Many  a  child  makes 
such  statements  almost  without  error  in  whose  mind 
there  is  nothing  back  of  the  words  that  are  uttered. 
The  child  who  reads  the  best  and  who  recites  without 
hesitation  may  be  the  very  one  who  does  not  really 
discern  the  relations  that  his  words  express.  To  recall 
words  and  sentences  and  repeat  them  glibly  is  one 
thing;  to  call  into  consciousness  supposedly  familiar 
magnitudes,  to  see  their  relation,  to  think  and  to 
express  thought,  is  quite  another.  The  first  of  these 
methods  of  recitation,  though  quite  satisfactory  to  the 
teacher  who  does  not  look  into  the  child's  mind,  gives 
responses  that  are  mathematically  and  pedagogically 
valueless.  The  second  calls  for  patient  waiting,  labo- 
rious imaging  and  comparing,  and  gives  at  first  tardy 
responses,  but  with  ever- increasing  power. 

Magnitude  and  Magnitude  Relation  to  be  in  the  Thought 
of  the  Child  when  Number  Symbols  are  Employed. 

The  one  unvarying  rule  in  this  early  work  is,  that 
magnitude  and  magnitude  relation  must  be  in  the 
thought  of  the  child  when  he  uses  the  number  sym- 
bols. If  he  says,  "i  half  of  5  is  2^,"  he  must  have 
in  mind  the  half  of  5  inches  or  5  apples  or  5  dollars  or 
5  hours — 5  in  the  concrete.     If  he  says,  "2  thirds  of 


ARITHMETIC:     HOW   TO   TEACH    IT  3 1 

6  are  4,"  he  must  have  in  mind  6  objects  divided  into 
3  equal  parts,  and  he  must  see  in  his  mental  picture 
the  2  twos  that  make  up  the  4.  If  he  says,  "6  is  (or 
are)  2  thirds  of  9,"  he  must  have  in  mind  6  objects 
divided  into  2  equal  parts,  and  then  as  many  objects 
added  to  these  as  there  are  in  one  of  the  parts. 

At  first,  of  course,  this  work  must  be  done  with 
the  actual  objects  present  to  the  senses.  But  soon, 
very  soon,  the  objects  must  be  concealed  from  view, 
and  the  teacher  must  see  that  their  images  are  present 
in  the  consciousness  of  the  child. 

The  Teacher  and  the  Mere  *' Hearer  of  Recitations" 
Compared. 

This    suggests   the    main   difference    between    the 

teacher  and  the  "hearer  of  recitations."     The  teacher 

is  ever  on  the  alert  to  induce  imaging  and 

The  Teacher, 
the  seeing  of  relation.    She  uses  objects,  but 

she  lays  them  aside  at  the  earliest  possible  moment. 
She  then  uses  the  symbols  for  these  (words  and  figures), 
and  watches  unceasingly  for  evidences  of  imaging 
power.  She  discovers  the  pupils  who  are  doing  by 
word  memory  what  they  should  do  by  seeing  magni- 
tude relation — who  are  trying  to  tell  relation  when  the 
things  related  are  not  in  consciousness.  These  she 
takes  back  to  the  sense  mgnitudes.  But  again  she 
quickly  conceals  the  objects  from  sight,  while  the 
pupils  repeat  their  efforts  to  see  them  with  the  mind's 
eye.  She  frames  her  questions  at  first  almost  wholly 
with  reference  to  testing  and  exercising  the  imaging 
power. 


32  ARITHMETIC:      HOW    TO    TEACH    IT 

The  "hearer  of  recitations,"  perhaps,  works  with 

objects — possibly    too    long — and    later    works    with 

symbols;    but   there  is   no   connection   be- 

The  "Hearer      ' 

of  Recita-       tween  the  object  work  and  the  symbol  work, 
ons.  yj^g  child  while  working  with  symbols  im- 

ages symbols,  and  not  that  for  which  the  symbols 
stand.  The  "hearer  of  recitations"  is  satisfied  if  the 
pupil  obtains  t}ie  answer,  and  in  explanation  uses  the 
language  that  is  in  accord  with  that  employed  by  one 
who  thinks. 

The  teacher  "goes  behind  the  returns."  She 
looks  into  the  child  mind.  She  asks  herself,  "Is  the 
The  Teacher,  pupil  obtaining  mere  figure  results  and  recit- 
ing mere  words,  or  is  his  work,  his  recitation,  the 
natural  expression  that  follows  the  discernment  of  re- 
lation— thinking?"  The  teacher  frequently  changes 
her  own  view-point,  and  by  skillful  questioning 
attempts  to  discover  the  real  background  of  the  child's 
expression.  If  he  images  and  sees  relation,  he  is 
encouraged  to  proceed.  If  his  mental  equipment 
is  mainly  word  and  figure  pictures,  she  takes  him  back 
again  to  the  things  of  sense. 

The  Primary  Facts  of  Number  must  be  Memorized. 

The  number  facts  given  on  pages  6  and  7  of  Book 
I.*  must  not  only  be  perceived  by  the  pupil;  they 
must  be  perfectly  memorized.  These  facts,  which 
have  come  to  him  through  discernment  of  magnitude 
relation,  must  now  be  generalized,  and  become  a  per- 
manent ^nemory  possession.      He  must  remember  that 

'Elementary  .Arithmetic,  pages  S  and  6. 


ARITHMETIC:     HOW    TO    TEACH    IT  33 

7  and  5  are  12 — always  12 — regardless  of  the  magni- 
tudes to  which  these  numbers  may  be  applied.  Such 
facts  must  now  come  to  him  by  a  pure  act  of  memory, 
without  the  expenditure  of  any  energy  in  imaging  and 
combining.  The  symbols,  7  +  5.  or  I,  or  5,  must  sug- 
gest 12  almost  as  quickly  as  the  figures  12  suggest  this 
number.  There  must  be  no  failure  in  the  little  memory 
work  that  is  necessary  to  the  complete  mastery  of 
pages  6  and  7  of  Book  I.  There  can  be  no  proper 
and  advantageous  committing  to  memory  until  the 
facts  are  discerned.  There  can  be  but  little  advance- 
ment until  the  facts  are  memorized.  Perceive — MEM- 
ORIZE. 

Werner  Arithmetic,  Book  I.,  Page  9.* 

Pages  6  and  7  having  been  mastered,  the  book  may 
be  put  into  the  hands  of  the  pupil ;  but  even  now  it 
should  be  closed  until  the  teacher  has  assured  herself 
that  the  pupils  are  familar  with  every  number  fact 
required  for  the  proper  reading  of  page  9.  To  do 
this,  the  teacher  should  read  the  page  aloud,  xeacher 
the  Dupils  filling  the  blanks.       If  there  is  Reads-Pupiis 

''  ,        .        .  f  .,       Listen,  and 

some  hesitation  on  the  part  of  many  pupils,  supply  Omit- 
this  work  should  be  several  times  repeated  t«^  Numbers, 
before  the  pupils'  books  are  opened.  The  number 
facts  on  the  page  having  been  mastered  and  reviewed, 
the  pupil  may  with  profit  review  them  again  by  reading 
the  page.  This  he  should  be  able  to  do  with  little 
hesitation.  If  he  has  been  properly  prepared  for  the 
work,  he  should  read  the  page  in   from  two  to  three 

♦Hall's  Elementary  Arithmetic,  page  9. 


34  ARITHMETIC:     HOW   TO    TEACH    IT 

minutes.  Thus  the  work  of  the  teacher  is  mainly  aid- 
ing the  pupil  to  prepare  himself  to  read  the  page. 

The  order  of  procedure  suggested  in  the  foregoing 
paragraph  may  be  applied  to  most  of  the  pages  of 
Book  I.  and  to  many  pages  of  Book  II.  This  method 
will  meet  the  approval  of  those  who  will  take  the 
trouble  to  test  it,  as  well  as  of  those  who  examine  it 
from  a  psychological  point  of  view. 

Eye  or  Ear? 

The  third-grade  pupil  has  already  acquired  a  good 
degree  of  skill  in  getting  thought  from  language 
addressed  to  the  ear.  He  has  had  a  comparatively 
small  amount  of  experience  in  getting  thought  from 
language  addressed  to  the  eye.  When  he  entered 
school  for  the  first  time,  he  had  listened  to  language 
for  about  six  years.  During  the  two  years  he  has 
been  in  school  he  has  heard  a  hundred  times  as  many 
Pupils  "Ear-  thought  symbols  as  he  has  seen.  For  this 
Minded,"       reason  he  has  vastly  more  skill  in  bringing 

so  Far  as  .  .  ^  .  **     ^ 

Language  is  into  consciousness  that  for  which  spoken 
Concerned.  symbols  Stand  than  he  has  that  which  is 
expressed  in  written  or  printed  symbols.  If  the  main 
task,  then,  is  to  image  and  to  see  relation,  the  prob- 
lem can  best  be  presented  by  addressing  the  ear. 
When  the  pupil  opens  the  book,  the  same  problems 
are  re-presented  by  means  of  printed  characters. 
While  the  work  was  being  done  orally,  the  task  of  the 
child  was,  (i)  hearing  symbols,  (2)  imaging,  and  (3) 
seeing  relation.  When  he  takes  the  book,  his  task  is, 
to  him  the  more  difficult  one:  (i)  seeing  symbols,  (2) 


ARITHMETIC:      HOW   TO    TEACH    IT  35 

imaging,  and  (3)  seeing  relation.  How  long  it  will 
be  necessary  that  he  shall  be  thus  assisted  by  the 
voice  of  the  teacher  depends  upon  the  skill  of  the 
teacher  and  the  ability  of  the  pupil.  The  amount  of 
oral  preparatory  work  may  be,  from  time  to  time, 
somewhat  diminished;  Sut  it  must  not  be  omitted 
until  the  pupil  has  acquired  the  power  easily  to  see  in 
the  printed  symbol  that  for  which  it  stands. 

This  method  of  procedure  is  a  part  of  essential 
gradation  in  all  the  branches  of  study  in  the  early 
years.  The  most  successful  teachers  of  reading  are 
those  who  observe  this  psychological  principle;  who 
"tell  the  story,"  or  at  least  a  part  of  it — enough  to 
arouse  the  interest  of  the  pupil,  and  to  put  into  opera- 
tion his  imaging  activity — before  the  reading  is 
attempted. 

The  "Spiral  Advancement"  Plan. 

Says  Le  Fevre:  "There  are  seven  distinct  numer- 
ical operations.  *  «  *  These  seven  operations 
are,  by  name,  addition  and  its  inverse,  subtraction ; 
multiplication  and  its  inverse,  division ;  involution 
and  its  two  inverses,  evolution  and  finding  the 
logarithm."* 

With  the  last  three  of  these  operations  the  pupil 
in  the  lower  grades  is  not  concerned.  Expanding  the 
fourth  (division)  into  two  thought  processes,  we  have, 
as  the  distinct  operations  of  arithmetic  for  the  elemen- 
tary school,  (i)  addition,  (2)  subtraction,  (3)  multipli- 
cation, (4)  division  (finding  how  many  times  one  num- 

*  Number  and  Its  Algebra,  page  42.— D.  C.  Heath  &  Co. 


36  ARITHMETIC:      HOW   TO    TEACH    IT 

ber  is  contained  in  another),  and  (5)  division  (finding 
a  certain  part  of  a  number). 

These  five  operations,  the  repetitions  of  which 
The  Eiemen-  "^vith  a  variety  of  magnitudes  constitute  the 
taiy  Spiral,  elementary  spiral  of  the  Werner  Arithme- 
tics, appear  in  regular  order,  five  times  on  page  9, 
twice  on  page  10,  five  times  on  page  11,  twice  on 
page  96,  and  more  than  one  hundred  and  fifty  times 
in  Book  I.  Each  time  a  turn  of  the  spiral  appears 
something  new  is  presented,  differing  only  slightly 
from  that  with  which  the  child  is  already  familiar. 
Small  At  first  the  numbers  are  small  and  the  prob- 

Kumbers.  ]ems  easy.  Very  gradually  greater  num- 
bers are  introduced.  (Compare  the  numbers  on  page 
9  with  those  on  any  subsequent  page — e.  g,,  pages 
II,  27,  41,  51,  61.)  In  some  parts  of  the  elementary 
"Abstract  spiral  the  so-called  abstract  numbers  are 
wnmbers."  employed;  others  are  made  up  of  five  or 
more  simple  problems  in  applied  arithmetic.  (Com- 
pare the  fifth  set  of  problems  on  page  9  or  1 1  with 
"Concrete  the  four  preceding  sets  on  the  same  page.) 
Numbers."  Gradually  denominate  numbers  in  consider- 
able variety  appear — e.  g.,  numbers  of  inches  and 
Denominate  f^^t  on  pages  ID  and  12;  numbers  of  pints, 
Numbers.  quarts,  and  gallons  on  pages  i6  and  47. 
At  first  integral  numbers  are  employed,  but  common 
Integral  fractions  are  early  introduced..     See  pages 

Numbers.        ^^^  55^   5-^    -r^^    e^c.      Decimals  appear  on 

Fractions.       pages  70,  80,  90,  and  as  a  turn  of  the  spiral 

Decimals.  on  pages   I55,    I57,    I95,    I97.* 

♦Hall's  Elementary  .Arithmetic,  pages  153,  155;  193,  195. 


ARITHMETIC:     IIOW   TO   T£ACH   IT  37 

Work  Outside  the  Elementary  Spiral. 
The  work  which  is  outside  the  formal  turns  of  the 
elementary  spiral  in  Book  I.  may  be  considered  under 
several  heads,  as  follows : 

1.  The  introduction  of  new  magnitudes  and  such 

reductions    (changes   in    form   of  expression)    as    are 

necessary  that   these   may  be   employed  in 

.  ■'  ^     ^  New  Magni- 

the  spiral — may  be  added,  subtracted,  mul-  tudes  and 
tiplied,  etc.      See  problems  i  to  4  on  pages     ^  "'^  ^°^^' 
45'  55»  75»  and  85;  and  pages  47  and  57. 

2.  Problems  for  practice  similar  to  those  in  the 
spiral,  but  designedly  presented  out  of  the  regular 
order.  Sometimes  a  single  problem  involves  uot  in  Reg- 
two  or  more  of  the  fundamental  operations,  ™^'  Q^^f>r. 
as  I  of  15  inches  are  —  inches;  15  inches  are  |  of 
how  many  inches?  See  page  52,  problems  15,  16, 
17;  page  62,  problems  14  to  17.  Sometimes  the 
problems  require  both  a  reduction  and  one  or  more  of 
the  fundamental  operations.  See  page  47,  problems 
14  to  20.* 

3.  The  introduction  of  new  number  facts  at  regu- 
lar intervals.      See   pages   41,   51,   61,   71,   wew  Num- 

81,   etc.  ber  Facts. 

4.  Exercises  to  test  and  cultivate  the  imaging 
power,  and  at  the  same  time  to  lead  the  pupil  to  see 
magnitude  and  magnitude  relation  in  figures,  cultivate  im- 
See  page  39;!  pages  49,  59,  69,  problems  aging  Power. 
4  to  6;  page  79.  Moreover,  the  sequence  of  problems 
is  such  as  will  induce  imaging.      See  page  41,  prob- 

*  Problems  10  and  11  in  Hall's  Elementary  Arithmetic, 
t  Hall's  Elementary  Arithmetic,  pages  33  and  34. 


38  ARITHMETIC:      HOW   TO   TEACH   IT 

lems  10,  II,  12,  etc.;  page  48,  problems  i  to  8;  page 
51,  problems  12,  13,  14,  etc.;  page  58,  problems 
I  to  12. 

5.  The  introduction  of  new  terms,  with  which  the 
pupil  is  expected  to  become  familiar  by  use  rather 

than    by    definition.      See    pages    53,    63, 

New  Terms, 

and  73. 

6.  Carefully  graded  exercises  in  figure  manipula- 

tion.     See  last   part   of  pages  44,   48,  50, 
Ciphering. 

54,  58,  60,  64,  68,  70,  etc.* 

Further  Explanation  of  the  Plan  of  Book  I. 

It  will  be  apparent  that  Book  I.  is  not  simply 
a  collection  of  graded  miscellaneous  problems,  but 
that  it  is  a  book  built  on  a  plan.  The  elementary 
spiral  is  at  once  the  foundation  upon  which,  and  the 
The  Central  Central  Column  around  which,  are  arranged 
Column.  jj^  somewhat  regular  order  such  exercises  as 

will  train  the  pupil  in  the  discernment  of  quantity 
relation,  and  prepare  him  for  the  gradual  ascent  of  the 
spiral.  At  every  turn  the  fact  is  impressed  upon  him 
that  the  work  involves  only  five  numerical  operations. 
(These  he  may  at  length  learn  to  reduce  to  four.) 
He  adds,  subtracts,  multiplies,  divides,  and  divides. 
New  magnitudes  appear,  but  no  new  process  appears, 
unless  it  be  that  of  changing  the  form  of  quantitative 
expression — reduction.  He  soon  discovers  that  he 
needs  only  to  become  familiar  with  a  new  kind  of 
magnitude  and  the  symbols  th^t  express  it,  learn  to 
change  the  form  of  these  expressions  of  quantity  with- 

*See  last  part  oi  each  page  ol  Hall's  Elementary  Arithmetic. 


ARITHMETIC:     HOW   TO   TEACH   IT  39 

out  changing  their  value,  and  he  can  immediately 
make  use  of  them  in  the  spiral,  and  move  forward  and 
upward  another  turn. 

It  will  be  observed  that  the  author  has  followed 
closely  the  recommendation  of  the  Conference  on 
Mathematics  appointed  by  the  Committee 

^f  .  ^  AGreatWum- 

of  Ten  in  providing  "a  greater  number  of  ber  oi  simple 
exercises  in  simple  calculation  and  in  the  ^"^"^®*- 
solution  of    concrete    problems"    than    have    usually 
appeared  in  text-books  for  the  third  and  fourth  grades. 
While  many  concrete  problems  are  given,  the  pupil  is 
not  allowed  to  lose  sight  of   the  possible  concreteness 
of  all  the  problems  given.     When   he  tells  Teuing  the 
the   meaning   of  an   abstract   problem   (see  Meaning, 
pages  157  and    197),*  he  thinks  magnitude  into  the 
problem.     When  he  tells  the  number  story  suggested 
by  the  probleny(see  pages  157  and  197),  he  xeiiing  the 
thinks  a  particular  kind  of  magnitude  into  Numberstory. 
the   problem.      He   is   not   permitted    to    forget    that 
figures  stand   for  magnitude  and  magnitude  relation. 
The  moment  he  is  in  danger  of  such  forgetfulness  he 
is  asked  to  tell  the  meaning — tell  the  number  story. 

"Mental"  or  "Written"? 

While  a  large  part  of  the  work  is  of  the  so-called 
mental  order,  there  are  hundreds  of  problems  for  the 
slate.  These  are  carefully  graded,  and  the  pupil  is 
expected  to  solve  many  of  them  "mentally"  before 
he  solves  them  with  the  aid  of  a  pencil.  See  note  at 
bottom  of  each  of  the  following  pages:  27,  28,  31, 

♦In  Hall's  Elementary  Arithmetic,  see  pages  155  and  195. 


40  ARITHMETIC:      HOW    TO    TEACH    IT 

32,  33,  34,  35.*  The  pupil  who  is  thus  taught  will 
properly  regard  the  pencil  as  a  convenience  in  the 
doing  of  that  which  it  is  quite  possible  for  him  to 
do  without  using  such  an  implement — even  without 
figures.  He  is  led  to  think  number  as  magnitude  and 
number  as  relation  often  enough  to  prevent  his  falling 
into  the  habit  of  figure  juggling. 

In  many  instances  a  problem  for  the  slate  is  pre- 
ceded by  a  similar  "mental"  problem.  This  plan  not 
A  "Mental"  o"ly  leads  the  pupil  to  the  solution  of  the 
Problem  Foi-   niore  difficult  problem,    but  it  insures  his 

lowed  by  a  1      •  ,     r  1 

Problem  for  grasp  of  the  number  relations  before  he 
the  Slate.  begins  the  figure  process.  This  feature  of 
the  Werner  Arithmetics  is  believed  to  be  new  and 
valuable.  See  pages  172,  173,  212,  227,  240,  247, 
248,  249.1 

Multiplying  by  a  Fraction. 

Great  emphasis  is  put  upon  the  importance  of  lead- 
ing the  pupil  to  observe  the  meaning  of  problems  in 
multiplication,  particularly  when  the  multiplier  is 
a  fraction  or  a  mixed  number.  See  page  114;  page 
126,  foot-note;  page  129,  problem  4,  foot-note;  page 
136,  foot-note;  page  146,  problem  4;  pages  165  and 
166;  page  175,  problems  14  and  15;  page  176,  prob- 
lem (c);  page  185,  problems  14  and  15;  pages  206, 
207,  and  208.$ 

*In  Hall's  Elementary  Arithmetic,  see  Suggestions  to  Teachers,  pages  242, 
243.  Note  especially  the  suggestions  with  relerence  to  the  work  on  pages  11, 
12,  13,  14,  and  15. 

fin  Hall's  Elementary  Arithmetic,  see  pages  165,  166,  167,  168. 

fin  Hall's  Elementary  .Arithmetic,  page  104,  and  hrst  three  notes  page  246. 


ARITHMETIC:     HOW   TO   TEACH    IT  41 

The  Double  Aspect  of  Division. 

The  two  thought  processes  in  division  are  made 
prominent  throughout  the  book.  The  author  does 
not  think  it  practicable  to  lead  the  child  to  see  mag- 
nitude and  magnitude  relation  in  the  figure  xwo  Thought 
manipulation  of  division  without  making  Processes. 
this  distinction.  The  psychologist  may  be  right  who 
regards  all  division  (in  the  sense  in  which  the  term 
is  employed  in  arithmetic)  as  the  process  of  finding 
how  many  times  one  number  is  contained  in  another; 
who  declares  that  we  find  one  fifth  of  a  number  by 
finding  how  many  times  five  is  contained  in  the  number. 
But  the  author  of  the  Werner  Arithmetics  is  unwilling 
to  concede  that  the  figure  process  of  division  can  be  well 
taught  without  presenting  the  subject  to  the  pupil  in 
its  double  aspect.  True,  when  required  to  "divide 
$2465  by  5,"  he  may  be  led  to  divide  2465  by  5,  or 
$2465  by  $5,  and  then  to  interpret  the  result.  But 
this  does  not  seem  to  be  the  direct  and  pedagogical 
method  of  approaching  the  subject.  It  is  immeasur- 
ably better  that  the  pupil  should  see,  before  he  per- 
forms the  figure  process,  that  if  he  is  required  to 
"divide  $2465  by  5,"  he  must  find  one  fifth  $2465  +  5. 
of  $2465;  and  if  he  is  required  to  "divide  «2465-^$5. 
$2465  by  $5,"  he  must  find  how  many  times  $5  is 
(or  are)  contained  in  $2465.  This  method  puts  the 
seeing  of  magnitude  relation  into  the  foreground,  and 
relegates  figure  manipulation  to  the  rear,  to  be 
brought  forward  whenever  it  is  needed  as  a  conve- 
nience in  seeing  relation.     The  importance  which  the 


42  ARITHMETIC:      HOW    TO   TEACH    IT 

author  attaches  to  the  making  of  this  distinction  in 
the  division  problems  may  be  seen  by  examining  the 
following  in  Book  I. :  *Page  1 5  ;  last  half  of  page  2 1 ; 
page  26,  problems  at  bottom  of  page,  and  foot-note; 
pages  36  and  37;  page  58,  problems  near  the  bottom 
of  page,  and  foot-note ;  page  60,  last  line  of  problems, 
and  foot-note ;  page  64,  problems  4  and  5 ;  page  70, 
last  line  of  problems,  and  foot-note;  page  74,  prob- 
lems 3  and  4;  page  78,  problems  near  bottom  of  page, 
and  foot-note ;  page  80,  last  line  of  problems,  and  foot- 
note; page  83,  problems  12  to  19,  and  foot-notes, 
etc.  It  will  be  observed  that  this  distinction  is  also 
made  in  the  last  two  processes  in  each  turn  of  the 
elementary  spiral. 

Division  of  Fractions. 
The  foregoing  method  is  essential  as  a  preparation 
for  the  study  of  division  of  common  and  decimal  frac- 
tions. If  the  young  pupil  is  expected  to  deal  with 
fractional  units,  these  must  be  approached  on  the 
magnitude  side,  otherwise  figure  juggling  will  be  the 
inevitable  result  of  the  effort.  Before  attempting  to 
divide  a  fraction  by  a  fraction,  a  fraction  by  an 
integer,  or  an  integer  by  a  fraction,  he  must  have 
a  clear  conception  of  what  he  wishes  to  do — what  he 
is  to  find.  A  pupil  may  be  taught  to  ^'Invert  the 
divisor  and  proceed  as  in  multiplication, ' '  and  be  utterly 
ignorant  of  the  relation  of  the  quantities  involved. 
He  can  get  the  answer;  and  when  a  few  years  later  he 

♦In  Hall's  Elementary  Arithmetic,  see  page  15;  page  21,  problems  11  to  14; 
page  30;  page  64,  problems  1,  2,  3,  4;  page  74,  problems  i,  2,  3,  4;  page  84,  prob- 
lems I,  2;  page  04,  problems  1, 2.  See  also  problems  at  bottom  of  pages  64  and  65, 
74  and  75,  84  and  85.  etg. 


ARITHMETIC:     MOW  TO  TEACH  IT  43 

is  required  to  "explain  division  of  fractions,"  he  can 
commit  to  memory  the  "explanation"  provided  by 
the  teacher.  Perhaps  it  never  occurs  to  the  teacher 
that  the  "explanation"  is  an  attempt  to  put  quantity 
and  quantity  relation  into  symbols  from  which  these 
should  never  have  been  allowed  to  escape.  The  pupil 
who  is  well  taught  will  not  for  any  great  length  of 
time  use  symbols  that  are  devoid  of  content.  True, 
when  quite  familiar  with  a  set  of  symbols  he  may  very 
properly  use  them  without  stopping  at  every  turn  to 
bring  into  consciousness  their  real  content;  but  he 
should  be  able  to  do  this  whenever  danger  of  obscurity 
makes  it  necessary.  He  should  always  be  able  to  see 
through  the  symbol  that  for  which  it  stands.  Inabil- 
ity on  the  part  of  the  pupil  to  "concrete"  a  figure 
problem — to  tell  its  meaning  before  solving  it,  and  to 
tell  a  suggested  number  story  after  solving  it — is  a  cer- 
tain indication  that  symbol  manipulation  is  coming 
too  rapidly  and  too  prominently  into  the  foreground 
of  the  child's  thoughts. 

There  are  three  kinds  of  problems  in  division  of 
fractions  that  need  especial  attention  with  respect  to 
their  meaning.     Examples  of  these  follow: 

I. 

$8^$!-=         8^-1=  %^^%\=         Ki= 

$4^$.5=        4-. 5=        $^^$2=         \^2  =  ,   etc. 

Any  one  of  the  above  problems  may  be  regarded  as 
meaning,  Find  how  many  times,  etc. ;  that  is,  each 
number  is  thought  of  as  a  magnitude,  the  problem 
being  to  find  the  ratio  of  the  first  to  the  second. 


44  ARITHMETIC:      HOW    TO   TEACH    IT 

2. 

$.8^43r        .8h-4=        $2.4-5-4=        2.4-^4=,  etc. 

Any  one  of  the  above  problems  may  be  regarded  as 
meaning,  Find  one  fourth  of,  etc. ;  that  is,  the  first 
number  stands  in  consciousness  for  magnitude,  the 
second  for  ratio.  The  problem  is  to  find  the  other 
magnitude. 

3- 
$6^^=         %\^\=         $15-^2^=         $275-^2.5  = 

Here  the  first  number  in  each  problem  stands  for 
magnitude  and  the  second  for  ratio.  The  problem, 
as  in  No.  2,  is  to  find  the  other  magnitude.  The  fact 
that  the  ratio  is  a  fraction  or  mixed  number  makes  it 
somewhat  more  difificult  of  interpretation.  The  pupils 
of  the  lower  grades  should  not  be  confronted  with 
such  problems  as  these.  For  the  method  of  approach 
to  the  third  variety  of  problems  in  division  of  fractions, 
see  Book  II.,  page  162.*  No  examples  of  this  kind 
appear  in  the  Werner  Arithmetics  on  any  page  pre- 
ceding the  one  mentioned  above. 

Werner  Arithmetic,  Book  Il.f 

Here  the  main  spiral  along  which  the  pupil  advances 
is  made  up  of  the  following  topics:  (i)  Simple  Num- 
bers, (2)  Common  Fractions,  (3)  Decimals,  (4)  Denom- 
inate   Numbers,    (5)    Measurements,    (6)     Ratio    and 

♦The  work  here  referred  to  is  not  given  in  the  Hall  Arithmetics  in  the  form 
in  which  it  is  given  in  the  Werner  Arithmetic,  Book  II. 

tThe  first  149  pages  of  Hall's  Complete  Arithmetic  are  nearly  identical 
with  the  first  14.9  pages  of  the  Werner  Arithmetic,  Book  II. 


ARITHMETIC:      HOW    TO   TEACH    IT  45 

Proportion,   (7)  Percentage,  (8)  Review,   and  (9)  Mis- 
cellaneous   Problems.     Two    pages   are    devoted    to 
percentage   and   one   page   to   each   of  the  a  Topic  is 
other  topics.      Thus  each  complete  turn  of  ^^-presented 

'^  t^  on  Every 

this  larger  spiral  occupies  exactly  ten  pages  Tenth  Page. 

of  the  book.     This  plan  is  adopted  as  a  convenience 

for  reference  and  for  review. 

In   passing   over  the   book   for  the  first  time  the 

pages  should   be  assigned   in  their  regular  order,  as 

in  any  other  book.       But  a  topic  may  be  pages  to  be 

reviewed  by  taking  every  tenth  page;  thus,    ^^^s°^^ 

if  after  passing  over  the  first  one  hundred  order. 

pages  of  the  book  the  class  seems  weak  in  the  work 

in  common  fractions,  review  pages  12,  22,  32,  42,  52, 

etc.      If  decimals  is  the  troublesome  subject,   „     ^ 

•'  Can  be 

review  pages   13.  23,  33,  43,    53,    etc.      If,   Reviewed 

for  instance,  page  127  seems  too  difficult  for    ^    °^**'' 

the  pupil,  review  page  117,  or  pages   107  and   117,  or 

pages  97,  107,  and  117.     This  plan  enables  the  teacher 

easily   to   select   those  parts  of  the  book  for  review 

which   are   in  the  direct  line  of  preparation   for  the 

special  difficulty  confronting  the  pupil. 

The  elementary  spiral  is  not  lost  sight  of  in  Book 

II.      It  appears,  either  in  part  or  complete,  on  many 

of    the    common-fraction    pages    and    upon  „^  ^, 

^   °  ^         The  Elemen- 

some  of  the  decimal  pages.      See  pages  72,   tary  spiral 
82,  92,  102,  and  problems  at  the  bottoms 
of  pages   13,   23,  33,  43,  53,   etc.     On   the   first   few 
common-fraction  pages  (12,  22,  32,  42)  multiplication 
is  omitted  from  the  spiral.     This  is  done  for  the  pur- 
pose of  throwing  together  and  putting  emphasis  upon 


46  ARITHMETIC:      HOW   TO   TEACH    IT 

these  processes  in  fractions  in  which  it  is  necessary  or 
convenient  to  operate  with  two  or  more  fractions  hav- 
ing Hke  denominators. 

Simple  Numbers. 

Under  this  head  the  pupil  is  gradually  made  familiar 
with  the  terms,  exact  divisor,  odd  number,  even  number ^ 
integral  number,  fractional  number,  mixed  number ,  prime 
number,  composite  number ,  factor ,  prime  factor ,  multiple, 
common  multiple,  and  least  common  multiple.  He  is 
taught  to  multiply  by  any  number  of  tens,  any  num- 
ber of  hundreds,  etc. ;  to  divide  by  any  number  of 
tens,  hundreds,  etc.  He  learns  the  meaning  of  the 
term  average,  and  solves  problems  involving  the  use 
of  simple  numbers  in  great  variety.  See  pages  ii, 
21,  31,  41,  51,  61,  etc. 

Common  Fractions. 
Under  this  head  the  pupil  is  taught  to  apply  the 
elementary  spiral  to  every  variety  of  problems  in  com- 
mon fractions.  By  a  frequent  requirement  to  "tell 
the  meaning"  and  to  "tell  the  suggested  number 
story,"  the  magnitude  idea  is  kept  prominent  in  the 
mind  of  the  pupil  while  he  learns  to  manipulate  frac- 
tion symbols.  "Three-story  fractions"  and  "mathe- 
matical monstrosities"  do  not  appear  in  the  book. 
All  the  processes  are  taught  with  fractions  having 
small  denominators.  Indeed,  only  such  fractions  as 
are  needed  in  the  solution  of  ordinary  business  prob- 
lems are  here  introduced.  See  pages  12,  22,  32,  42, 
52,  62,  etc. 


ARITHMETIC:     HOW    TO   TEACH    IT  47 

Decimals. 

Here  again  the  elementary  spiral  appears;  but  at 
first  no  fractions  with  denominators  greater  than  1000 
are  introduced.  As  in  common  fractions,  the  magni- 
tude idea  is  made  prominent.  Here  again  the  pupil 
is  asked  to  "tell  the  meaning."  By  this  plan  the 
child  knows  how  to  "point  off"  his  answer  before  he 
begins  the  figure  process.  He  is  not  allowed  to  mul- 
tiply, for  instance,  by  .4  until  he  knows  that  this 
means,  find  /j.  tenths  of  the  multiplicand.  To  find 
4  tenths,  he  must  first  find  i  tenth.  He  discerns 
before  he  begins  the  process  of  multiplication  that  he 
is  simply  to  find  4  times  i  tenth  of  the  multiplicand. 

He  is  not  allowed  to  divide  $.385  by  $.005  until  he 
knows  that  he  is  to  find  how  many  times  5  thousandths 
are  contained  in  385  thousandths.  Knowing  this,  his 
figuring  is  in  no  way  different  from  that  necessary  to 
find  how  many  times  5  bushels  are  contained  in  385 
bushels.  He  is  not  allowed  to  divide  $.385  by  5  until 
he  knows  that  he  is  to  find  i  fifth  of  $.385.  Knowing 
this,  his  figure  work  is  in  no  respect  different  from 
that  necessary  to  find  i  fifth  of  385  bushels. 

A  problem  having  been  solved,  the  pupil  may  be 
required  to  "tell  the  suggested  number  story" — that 
is,  to  give  an  example  from  real  or  prospective  experi- 
ence in  which  the  given  figure  process  may  be  em- 
ployed To  make  this  possible,  no  unreal  or  imprac- 
ticable problems  are  introduced. 

N6t  a  problem  appears  in  Book  H.  that  does  not 
have  its  easily   found  parallel  in  the  outside  world, 


48  ARITHMETIC:      HOW    TO    TEACH    IT 

usually  very  close  to,  if  not  a  part  of,  the  child's  experi- 
ence or  observation.  See  pages  13,  23,  33,  43,  53, 
63,  etc. 

Denominate  Numbers. 

Instead  of  presenting  the  denominate  number  tables 
to  be  committed  to  memory  in  two  or  three  lessons, 
the  different  units  of  measurement  are  gradually  intro- 
duced into  the  problems  given  under  this  head.  Thus, 
on  page  14,  we  have  problems  involving  dollars,  tons, 
an^  pounds:  on  page  24,  bushels,  tons,  and  pounds; 
on  page  54,  feet,  rods,  and  miles;  on  page  74,  feet, 
rods,  yards,  inches,  miles,  tons,  pounds,  dollars,  and 
cents.  Moreover,  on  these  pages,  and  on  the*  review 
and  miscellaneous  pages  (19  and  20,  29  and  30,  39 
and  40,  etc.),  the  units  of  measurement  that  were 
treated  in  Book  I.  are  kept  constantly  before  the  pupil. 
The  spiral  advancement  plan,  giving  opportunity  as 
it  does  for  a  most  systematic,  frequent,  and  thorough 
review,  enables  the  teacher  to  see  that  the  neiv  is 
always  related  to  the  old  in  the  mind  of  the  pupil. 
See  pages  14,  24,  34,  44,  54,  64,  etc. 

Measurements. 

All  arithmetic  has  to  do  with  measurement.  As 
has  been  said,  the  number  idea  originates  in  measure- 
ment, and  in  the  end  the  number  processes  are  applied 
to  measurement.  But  there  are  special  phases  of  this 
work  which  may  properly  be  considered  under  this 
head,  the  word  vicasiirenicnts  being  here  used  in  a 
restricted  sense. 

Linear,    surface,    and   solid   measurements  are  the 


ARITHMETIC:      HOW   TO    TEACH   IT  49 

subdivisions  of  this  topic.  The  problems  are  espe- 
cially adapted  to  the  training  and  development  of  the 
imaging  power.  Indeed,  length,  area,  and  volume 
are  the  extension  elements  of  all  visual  and  motor 
images.  Even  weight,  intensity,  value,  or  tempera- 
ture may  be  thought  of  (imaged)  as  length,  area,  or 
volume.  Hence,  by  far  the  greater  number  of  our 
images  involve  these  extension  elements.  No  work 
in  mathematics  can  be  more  important  than  the  train- 
ing of  the  pupil  to  bring  easily  into  consciousness 
images  of  these  kinds  of  magnitude. 

Here,  as  elsewhere,  sequence  has  much  to  do  with 
the  training  value  of  a  set  of  problems.  If  each  of 
twenty  consecutive  problems  calls  for  the  area  of 
a  square,  and  the  figure  process  of  each  is  quite  long, 
there  will  be  little  imaging  and  much  figure  manipula- 
tion on  the  part  of  the  pupil  solving  them.  If  prob- 
lem I  calls  for  area,  problem  2  for  perimeter,  problem 
3  for  area,  problem  4  for  perimeter,  etc. ;  or  problem 
I  deals  with  a  square,  problem  2  with  a  cube,  prob- 
lem 3  with  a  square,  problem  4  with  a  cube,  etc.,  and 
if  the  figure  manipulation  is  easy,  the  emphasis  will 
be  upon  imaging.  This  is  as  it  should  be;  for,  as  has 
been  said  before,  the  difficult  task  is  not  in  seeing 
magnitude  relation,  but  in  imaging  the  related  magni- 
tudes.    See  pages  15,  25,  35,  45,  55,  65,  etc. 

Ratio  and  Proportion. 

Here  the  pupil  is  especially  exercised  in  seeing 
number  as  magnitude  and  number  as  relation.  Care- 
ful attention  to  the  sequence  of  problems  again  puts 


50  ARITHMETIC:      HOW   TO    TEACH    IT 

the  emphasis  upon  imaging  and  seeing  relation.  ^^  One 
fourtli  of  20  is — ,  20  is  \  of — ,"  is  a  better  sequence 
for  training  in  seeing  relation  than,  ''  One  fourth  of 
20  is  — ,  one  fourth  of  B/f  is  — . "  Note  the  sequence 
of  problems  on  pages  i6,  26,  36,  46,  56,  66,  etc. ;  also 
on  pages  136  and  146. 

Percentage. 

It  is  believed  that  the  subject  of  percentage  as 
found  on  pages  17  and  18,  27  and  28,  37  and  38,  47 
and  48,  etc.,  is  arranged  in  such  a  step-by-step  order 
that  no  pupil  who  is  properly  prepared  to  undertake 
the  work  will  find  serious  difficulty  in  its  mastery. 
With  very  little  assistance,  the  pupil  should  be  able 
to  take  these  steps.  Page  17  is  preparation  for  page 
18;  and  pages  17  and  18  are  preparation  for  pages  27 
and  28.  The  lessons  should  be  assigned  in  the  regular 
order  of  the  pages,  as  in  any  other  book;  but  if  the 
percentage  problems  on  any  page  are  found  too  diffi- 
cult for  the  pupil,  he  need  not  necessarily  be  given 
assistance,  but  simply  directed  to  review  the  percent- 
age problems  on  the  pages  preceding  that  on  which 
the  difficulty  appears.  Indeed,  this  might  properly 
be  given  as  a  very  general  direction.  It  is  not  best 
that  the  teacher  should  solve  problems  for  the  pupil, 
or  show  him  how  to  solve  them.  If  they  are  too 
heavy  for  him,  give  him  those  which  he  can  solve,  and 
so  lead  up  to  and  over  the  difficulty. 


ARITHMETIC:     HOW   TO   TEACH    IT  5 1 

Werner  Arithmetic,  Book  III.* 

No  arithmetic  built  on  the  spiral  advancement  plan 
can  be  complete  unless  it  at  length  leads  the  pupil  to 
that  point  from  which  he  may  be  able  to  survey  the 
subject  as  a  whole,  and  assists  him  in  the  doing  of  this. 
Hence,  classification  and  generalization  are  the  features 
of  Book  III.  But  here,  as  in  Book  II.,  in  answer  to 
a  very  general  demand,  the  number  of  topics  treated 
is  by  no  means  so  great  as  the  number  found  in  the 
arithmetics  of  twenty  years  ago. 

The  fundamental  operations  in  their  application  to 
simple  numbers,  decimals,  United  States  money,  de- 
nominate numbers,  and  literal  quantities  are  treated 
briefly  under  the  four  general  heads — addition,  sub- 
traction, multiplication,  and  division.  The  other 
topics  presented  are,  properties  of  numbers,  common 
fractions,  percentage  and  its  applications,  ratio  and 
proportion,  powers  and  roots,  and  the  metric  system; 
to  which  are  added  a  special  chapter  (page  23 if)  on 
denominate  numbers,  one  on  short  methods,  and  many 
practical  problems. 

The  spiral  advancement  plan  is  followed  in  the 
gradual  introduction  of  elementary  work  in  algebra 
and  geometry.  See  pages  17,  18,  19;  27,  28,  29; 
37,  38,  39;  47,  48,  49,  etc.t  The  algebra  work,  espe- 
cially, is  closely  correlated  with  the  work  in  arithmetic. 
Six  pages  treating  of  figure  notation  are  followed  by 
two  pages  of  algebraic  notation ;  six  pages  treating  of 

*0r  Complete  Arithmetic,  Part  11. 

t Complete  Arithmetic,  page  371. 

jComplete  Arithmetic,  pages  157.  158,  159;  167,  t6R,  169:  i77.  «78i  iW.  etc. 


52  ARITHMETIC:      HOW   TO    TEACH    IT 

addition  as  it  appears  in  arithmetic  are  followed  by 
two  pages  of  algebraic  addition,  etc.  In  many  parts 
of  the  book,  particularly  in  fractions  (see  pages  87, 
88,  97,  98*)  and  in  proportion  (see  pages  197  and 
198!),  the  literal  notation  is  an  invaluable  aid  in  the 
generalization  of  the  work  in  arithmetic. 

Problems  which  "perplex  and  exhaust  the  pupil 
without  affording  any  really  valuable  mental  disci- 
pline" have  been  omitted. 

The  Werner  Series  of  Arithmetics. 

In  the  series  as  a  whole,  the  author  has  attempted 
to  "abridge  and  enrich"  the  subject  of  arithmetic. 
The  Abridg-  The  abridgment  consists  in  omitting  much 
ment.  ^j^g^^  j^^g  heretofore  been  regarded  as  essen- 

tial, particularly  work  involving  long  and  difficult 
figure  processes  that  have  no  parallel  in  ordinary  busi- 
ness arithmetic,  and  but  little  if  any  disciplinary  value. 
Such  phases  of  commercial  arithmetic  as  are  unintel- 
ligible to  the  average  child  of  twelve  or  fourteen  years 
are  also  omitted. 

The  enrichment  consists  in  the  introduction  of 
a  large  number  of  simple,  concrete  problems,  all  of 
The  Enrich-  which  have  their  parallel  in  the  environment 
ment.  ^f  ^^le  average  pupil. 

The  entire  series  presents  over  fifteen  thousand 
problems.  Those  in  Book  I.  are  arranged  into  and 
around  the  elementary  spiral — addition,  subtraction. 
Book  I.  multiplication,  division,  and  division.      Into 

this  spiral  at  somewhat  regular  intervals  appear  easy 

*  Complete  Arithmetic,  pages  227,  228;  237,  238. 
tComplete  Arithmetic,  pages  337,  338. 


ARITHMETIC:     HOW   TO   TEACH   IT  53 

problems  with  integral  numbers,  common  fractions, 
decimals,  United  States  money,  and  denominate  num- 
bers. 

The  problems  in  Book  II.  are  arranged  into  the 
larger  spiral  made  up  of  seven  topics;  namely,  simple 
Book  n.  numbers,  common  fractions,  decimals, 
denominate  numbers,  measurements,  ratio  and  pro- 
portion, and  percentage.  Each  turn  of  the  spiral 
covers  ten  pages  of  the  book. 

The  problems  in   Book  III.    lead   the  pupil  into 
a  survey  of  the  subject  as  a  whole,  with  its  usual  divi- 
sions and  subdivisions,  all  of  which  are  sup-  Book  m. 
plemented   by   elementary    exercises    in    algebra   and 
geometry. 

The  Hall  Arithmetics. 

The  Elementary  Arithmetic  is  substantially  the 
same  as  the  Werner  Arithmetic,  Book  I. 

The  Complete  Arithmetic  is  made  up  of  Werner 
Books  II.  and  III.  somewhat  abridged. 

The  Hall  Arithmetics  are  made  on  the  same 
general  plan  as  the  Werner  Arithmetics.  They  are 
designed  for  those  who  desire  a  somewhat  briefer 
course  than  that  provided  by  the  three-book  «eries. 


The  author  and  publishers  cannot  be  otherwise 
than  delighted  with  the  reception  that  has  already 
been  accorded  in  the  East  and  in  the  West  to  "The 
New  Arithmetics."  They  were  wrought  out  in  the 
schoolroom,  because  the  writer  felt  the  need  of  them 


54  ARITHMETIC:     HOW   TO   TEACH   IT 

in  his  own  work.  They  are  the  outgrowth  of  more 
than  a  third  of  a  century's  experience  in  presenting 
this  subject  to  pupils  and  to  teachers.  If  now  they 
shall  prove  in  a  large  degree  helpful  to  the  thousands 
of  teachers  and  tens  of  thousands  of  pupils  into  whose 
hands  they  have  already  found  their  way,  it  will  be 
a  source  of  immeasurable  satisfaction  to  their  author. 


It  is  difficult  to  complete  this  little  monograph 
without  the  introduction  of  the  personal  pronoun  of 
the  first  person.  In  thanking  one's  personal  friends, 
good  form  does  not  demand  the  pretended,  conceal- 
ment of  one's  own  personality  behind  the  author,  or 
by  the  use,  even,  of  the  editorial  we. 

I  beg  to  thank  most  sincerely  the  hundreds  (I  might 
almost  say  thousands)  of  friends,  many  of  whose  faces 
I  have  never  seen,  who  have  taken  the  trouble  to 
express  in  no  doubtful  terms  their  appreciation  of  the 
results  of  my  endeavor  to  "abridge  and  enrich"  the 
work  in  arithmetic  for  the  common  schools. 

I  desire  especially  to  express  my  gratitude  to  my 
good  friend  Orville  T.  Bright,  of  Chicago,  to  whom 
I  am  indebted,  more  than  to  any  other  teacher  or 
superintendent,  for  early  recognition  of  the  value  of 
my  work,  and  for  timely  assistance  in  bringing  it  to  the 
attention  of  an  enterprising  publishing  house,  whose 
services  have  already  made  me,  in  a  sense,  the  teacher 
of  MORE  THAN  A  QUARTER-MILLION  PUPILS. 

F.   H.  H. 
Jacksonville,  Illinois,  May.  1900. 


The  Werner  Arithmetics  tn  the  School-room 


The  Longest  Possible  Test 


DeKalb,  Illinois,  June,  1900. 

In  the  summer  of  1896,  while  superintendent  of  the 
schools  of  Austin,  Illinois,  1  received  a  copy  of  the  Werner 
Arithmetic,  Book  1.  This  book  had  at  that  time  been  before 
the  public  less  than  one  month.  1  had  during  the  previous 
year  observed  the  work  of  the  author  in  the  Waukegan 
schools,  and  was  in  sympathy  with  it.  1  expected  a  book 
aggressively  devoted  to  meeting  the  essential  needs  of  the 
grades  for  which  it  was  intended,  and  meeting  them  fully; 
a  book  adapted  by  its  clear  discernment  of  what  matter  to 
present,  of  the  order  and  rate  of  its  presentation,  by  its  stead- 
fast adherence  to  the  main  line  of  its  purpose,  by  the  ingenu- 
ity and  simplicity  of  its  method  and  arrangement,  and  by  its 
sharp  self-limitation  of  range— adapted  thus  unfailingly  to 
reach  its  end.- 

1  was  not  dissapointed.  The  book  went  beyond  my 
measure  for  it.  The  unity  was  very  marked;  the  holding  of 
the  several  lines  of  thought,  in  clear,  close  relationship,  un- 
slackened.  I  was  convinced  that  it  was  almost,  if  not  quite, 
the  ideal  book  for  its  place  in  the  curriculum. 

In  the  autumn  of  1896,  we  gave  Book  I  a  trial  in  our 
schools.  At  an  early  date  it  was  adopted  for  use  in  all  the 
grades  for  which  it  was  designed.  When  the  other  books  of 
the  series  appeared  (Book  II  In  1897,  and  Book  III  in  1898), 
these,  too,  were  adopted.  I  have  thus  had  continuous  experi- 
ence with  the  Werner  Arithmetics  as  long  as  it  is  possible  for 
any  one  to  have  had  such  experience. 

From  my  own  observation,  and  from  the  unanimous 
sentiment  of  my  teachers,  I  am  led  to  reaffirm  the  opin- 
ions early  formed  and  expressed.  Intelligent  fidelity  to 
the  author's  plan  results  in  stronger  grasp,  clearer  insight, 
greater  facility,  and  more  zest  in  dealing  with  numbers. 
The  teacher  is  helped  even  more  than  the  pupil. 

{Signed)  NEWELL  D.  GILBERT, 

Supt.  DeKalb  Public  Schools. 

Formerly  Supt.  Austin  Public  Schools. 


Bearborn  School 


Boston,  Mass.,  January  26,  igoo. 
Werner  School  Book  Company. 

Dear  Sirs: — The  Werner  Arithmetics  by  Frank  H. 
Hall  meet  my  hearty  approval  because  of  their  simplicity 
and  wise  classification.  They  embody  Prof.  Hall's  belief 
that  number  has  two  aspects,  magnitude  and  ratio.  These 
books  also  make  a  close  connection  between  mental  arith- 
metic and  written  arithmetic;  they  require  the  pupil  to  do 
his  own  thinking  and  make  his  own  rules.  The  problems 
grow  more  and  more  difficult,  but  they  change  gradually. 

These  books  call  for  "the  story,"  in  which  the  child, 
and  not  the  teacher,  makes  up  a  variety  of  examples;  re- 
views are  regularly  introduced,  and  drill  is  never  lost  sight 
of.  In  short,  these  books  seem  to  be  made  by  one  who 
understands  a  child's  mind,  and  is  anxious  to  do  what  is 
best  for  the  development  of  that  mind. 

My  teachers  are  highly  pleased  with  the  books. 
Yours  truly, 

C.  F.  King. 


Zbc  public  Scbools 

Citie  of  Beatrice 

Beatrice,  Ylebraslia 


Beatrice,  Neb.,  March  31,  1900. 
Werner  School  Book  Company, 
Chicago,  Illinois. 

Dear  Si'rs : — We  adopted  the  Werner  Arithmetics  in 
our  schools  last  August,  and  have  been  watching  the  results 
very  closely.  They  were  chosen  from  all  other  arithmetics 
because  we  regarded  them  as  the  best  published,  and  they 
have  fully  come  up  to  our  expectations. 

In  the  first  place,  arithmetic  is  no  longer  the  meaningless 
mass  of  abstract  computations  that  have  in  the  past  been 
the  bugbear  of  both  pupils  and  teacher.  The  Werner 
Arithmetics  make  both  study  and  teaching  a  delight. 

In  the  second  place,  the  pupils  have  their  mathematical 
powers  developed,  and  at  the  same  time  learn  to  apply 
these  powers  along  practical  lines,  and  acquire  a  knowledge 
of  numbers  that  will  always  be  of  use  to  them.  The  books 
are  also  valuable  for  the  omission  of  many  unnecessary 
subjects  that  have  cumbered  our  text-books  in  the  past. 

You  are  to  be  congratulated  upon  the  publication  of 
these  excellent  books.    Progressive  teachers  will  welcome 
them  wherever  they  can  be  introduced. 
Very  truly  yours, 

J.  W.  DiNSMORE, 

Superintendent  of  Schools. 


9(Bce  of 

^be  3Boar^  of  Je^ucation 

21  Center  Street 
flew  f)aven,  Connecticut 


New  Haven,  Conn. 
Werner  School  Book  Company: 

After  the  examination  of  various  text-books,  the  super- 
visory staff,  by  a  vote  of  eight  out  of  the  ten,  recommended 
the  Werner  Arithmetic,  Book  III,  and  it  was  unanimously 
adopted  by  the  Board  of  Education.  Among  the  reasons 
given  for  their  choice  were  these: 

"  The  book  omits  superfluous  subjects  in  Arithmetic." 
"  The  exerci:5es  are  more  nearly  like  the  combinations 
of  business  than  other  books,  therefore  the  tone  of  the  book 
is  practical,  and  it  is  believed   it  will   appeal   strongly  to 
pupils." 

"The  Algebra  and  Geometry  exercises  supplement 
the  Arithmetic  exercises,  and  are  not  simply  additions  to 
the  text." 

"  The  book  not  only  compels  thought,  but  induces  it." 
"  Mensuration  is  carried  through  the  book   in   a   most 
admirable  way." 

"The  admirable  page  arrangement  of  the  book  makes 
it  simple  and  plain  for  teachers  to  use." 

"  The  arrangement  of  the  book  is  ideal." 
Personally,  I  believe  the  Werner  book  the  most  rational 
contribution  that  has  been  made  to  arithmetical  text-books 
for  the  higher  grammar  grades. 

Yours  sincerely, 

C.  N.  Kendall, 

Superintendent  ot  Schools. 


'Tar he  IPs   Geographical  Series 

By  HORACE  S.  TARBELL,  A.  M.,  LL.  D. 

Superintendent  of  Schools,  Providence,  R.  I. 

And  MARTHA  TARBELL,  Ph.  D. 


A  TWO-BOOK  COURSE  FOR  GRADED 
AND  UNGRADED  SCHOOLS 


Profuse  with  Mapsy  Diagrams^  and  Illustrations 
prepared  expressly  for  this  work. 


Tarbeirs  Introductory  Geography 

Revised  to  date.  24  maps.  360  Illustrations. 
Small  Quarto.     Cloth,   188  pages.     50  cents. 

Tarbell's  Complete  Geography 

New  Text.  New  Maps.  New  Illustrations. 
The  Geography  of  the  World  down  to  the 
latest  date,  including  the  Foreign  Possessions  of 
the  United  States.  Large  Quarto.  Cloth,  152 
pages.     ;^i.oo. 


Correspondence  in  regard  to  the  examination  of 
Tarbell's  Geographies  and  their  introduction  into 
schools  is  cordially  invited. 

Werner  School  Book  Company 

EDUCATIONAL  PUBLISHERS 
NEW  YORK  CHICAGO  BOSTON 


TRAINING    FOR    CITIZENSHIP 

THE  FOUR 
GREAT  AMERICANS  SERIES 


BIOGRAPHICAL  STORIES  OF 
GREAT  AMERICANS 

EDITED  BY  JAMES  BALDWIN,  PH.  D. 

About  256  pages  each.   Cloth.    Illustrated.   Price  50  Cents  the 

Volume.    Published  also  in  flexible  Cover  Booklets, 

64  Pages  Each.     Price  10  Cents  per  Booklet. 

"THESE  Stories  present  the  lives  of  our  Great  Americans  in 
such  a  manner  as  to  hold  the  attention  of  the  youngest 
readers.  The  boyhood  of  each  of  these  great  men  is  described 
with  many  interesting  details.  The  manner  also  in  which 
each  educated  himself  and  prepared  for  his  life  work  has 
been  especially  dwelt  upon.  For  the  school  or  for  the  home, 
these  books  are  unique  and  valuable,  and  cannot  fail  to  have 
an  uplifting  influence  on  the  youth  of  America. 

As  no  books  have  done  before,  these  Life  Stories  serve 
the  following  purposes:  They  lay  the  foundation  for  the 
study  of  BIOGRAPHY  and  HISTORY:  they  stimulate  a  desire 
for  further  HISTORICAL  READING;  they  cultivate  a  taste  for 
the  Best  literature;  and  by  inspiring  examples  they 
teach  Patriotism. 

SEVERAL  VOLUMES  ALREADY  PUBLISHED 
OTHERS  IN  PRESS  ::  SEND  FOR  LIST 
Liberal  Terms  for  Supplies  to  Schools 

ADDRESS  the  PUBLISHERS 

Werner  School  Book  Company 

New  York        Chicago        Boston 


THE    "NEW    PENMANSHIP" 

RAPID  VERTICAL  WRITING 


The  Rational  Writing 
Books 

A  new  series  of  writing  books,  combining  in  their 
style  of  copies  and  system  of  practice  Legibility, 
Speed,  Beauty,  Simplicity  and  Ease  of  Execution, 
Economy  of  Time  and  Expense. 

RAPIDITY 

Owing  to  the  fact  that  vertical  writing  has  hereto- 
fore tended  to  slowness  of  movement,  the  subject  of 
rapidity  has  received  special  attention  in  these  books. 
This  great  objection,  sometimes  urged  against  vertical 
writing,  has  been  overcome  in  the  Rational  Writing 
Books. 

The  Rational  Writing  Books 

SIX  NUMBERS.     72  CENTS  PER  DOZEN 

Sample  copies  by  mail,  prepaid,  on  receipt  of  price. 
Correspondence  cordially  invited. 


Werner   School   Book   Company 

NEW  YORK.  CHICAGO  BOSTON 


The  Practical  Series  of 
School  Physiologies^  w.e. Baldwin, iw. d. 

With  Special  Reference  to  Hygiene,  and 
to  State  Laws  regarding  the  Nature  of 
Alcoholic  Drinks  and  other  Narcotics. 


Primary  Lessons  in  Human  Physiology  and  Hygiene 

The  Children's  Health  Reader. 

For  the  Fourth  Year  of  Graded  School  work;  and 
for  corresponding  classes  in  Ungraded  Schools, 
Extra  Cloth,  144  Pages.    Price  35  Cents. 


Essential  Lessons  in  Human  Physiology  and  Hygiene 

For  Intermediate  and  Grammar  Grades;  and  for 
corresponding  classes  in  Ungraded  Schools. 
Extra  Cloth,  200  Pages.    Price  50  Cents. 


Advanced  Lessons  in  Human  Physiology  and  Hygiene 

For  High  Schools  and  Advanced  Grammar  Grades; 
Normal  Schools,    Academies;  and  for  Advanced 
Classes  in  Ungraded  Schools. 
Extra  Cloth,  400  Pages.    Price  80  Cents. 


Send  for  Special  Circulars,  Terms,  Etc. 
WERNER  SCHOOL  BOOK  COMPANY, 

Educational  Publishers, 

New  York.  Chicago.  Boston. 


The  DeGarmo  Language 
—  Series  == 

A  COURSE  IN  ENGLISH  FROM 
PRIMARY  GRADES  TO  HIGH    SCHOOL 

LANGUAGE    LESSONS 

By  CHARLES  DE  GARMO,  Ph.  D. 

Professor  of  the  Science  and  Art  of  Education,  Cornell  University,  Ithaca,  N.  Y. 

ELEMENTS  OF 
ENGLISH  GRAMMAR 

By  GEORGE  P.  BROWN 

Former  Superintendent  of  Schools,  Indianapolis,  Ind. 

Assisted  by  CHARLES    DE   GARMO 

PRICE  LIST 

Language  Lessons,  Book  One,  145  pages    -    -    -    30  cents 
Language  Lessons,  Book  Two,  188  pages    -    -    -    40  cents 

Beautifully  and  Copiously  Illustrated. 

Complete  Language   Lessons,  in  one  volume,  256 

pages,  over  200  illustrations 50  cents 

Elements  of  English  Grammar,  256  pages      -    -    60  cents 

Send  for  Dkscriptivk  Circulars  of  our  Up-to-Date 

Epoch- Making  Text-Books 

Samples  to  Teachers  for  examination  at  special  rates. 
Liberal  terms  for  introduction  and  exchange. 


Two  Epoch -Makers 

The  Werner  Primer 

For  Beginners  in    Reading 

By  Frances  Lilian  Taylor 

Richly  illustrated  with  handsomely  executed  col- 
ored lithographs,  requiring  from  six  to  ten  distinct 
impressions.  The  most  revolutionary  school  book 
ever  published. 

112  Pages,  8vo.     Price,  30  Cents. 


The  First-Tear  Nature 

Reader 

By  Katherine  Beebe    and  Nellie  F.  Kingsley 

154  pages.      Price,  35  Cents. 

Illustrated  in  colors,  bound  in  delicate  green 
cloth,  with  artistic  cover  design. 


Werner  School  Book  Company 

EDUCATIONAL  PUBLISHERS 
NEW  YORK  CHICAGO  BOSTON 


R-^^* 


)  - 


^  000937J23 


^s^. 


*r  v^?!'-^,^'' 


,„tif. 


